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Cambridge University Press 978-0-521-88762-5 - Hilbert Transforms, Volume 1 Frederick W. King Index More information

Author index

Only a limited number of citations are included for Hilbert, Cauchy, and Fourier, as these authors occur very prominently throughout the book. Authors cited implicitly (as part of et al.) are indicated by italic font. Page numbers for volume 2 are indicated by (2).

Abdelouhab, L., 633 Ansari, R., (2)178 Abel, N. H., 1, 296, 537 Anton, J., (2)116 Ablowitz, M. J., 633 Arcozzi, N., 248 Abragam, A., (2)303, (2)450 Argand, J. R., 24 Abraham, M., (2)93 Arikan, O., (2)406 Abramowitz, M., 178, 268, 269, 425, 431, 453, Arm, M., (2)41 454, 457, 461, 463, 471, 522 Arnison, M. R., 8 Achenbach, J. D., (2)446, (2)450 Arocena, R., 414 Achieser, N. I., see Akhiezer, N. I. Artiaga, L., (2)70 Adams, E., 416 Asakura, T., (2)1, (2)406, (2)303, (2)396, (2)403 Afshar, R., 732 Askey, R., 570 Agneni, A., (2)116 Asmar, N., 211, 328 Agrawal, A., 681, (2)178 Aspnes, D. E., (2)249 Aheizer, N. I., see Akhiezer, N. I. Assani, I., (2)67, (2)69 Ahrenkiel, R. K., (2)303 Astala, K., 416 Akhiezer, N. I., 200, 201, (2)249 Atencia, E., (2)69 Aki, K., 8 Atkinson. K., 730 Akilov, G. P., 78, (2)42 Au, C.-K., 8 Alavi-Sereshki, M. M., 8 Audoin, B., (2)450 Albrecht, A. C., 8, (2)449 Alexander, R., (2)303 Babenko, K. I., 246, 359, 414, (2)45 Alfimov, G. L., 633 Baburin, O. V., 732 Alieva, T., (2)179 Bachmann, P., 11, 78 Almeida, L. B., (2)180 Baernstein, II., A., 337 Altarelli, M., (2)224, (2)241, (2)242, (2)243, Baird, L. C., 79 (2)248, (2)249, (2)250, (2)251, (2)304 Baker, G. R., 633 Alzer, H., 665 Balcou, Ph., (2)358 Amari, S., 731 Balzarotti, A., (2)303 Amusia, M. Ya., (2)450 Bambini, A., (2)248 Andermann, G., (2)303 Bampi, F., 8 Andersen, K. F., 328, 400, 401, 402, 414, 416, Bañuelos, R. (2)42 680, 681, (2)69 Bardaro, C., 681 Anderson, B. D. O., (2)179 Bareiss, E. H., 730 Anderson, D. G., 704 Barnes, A. E., (2)179 Anderson, M. A., (2)303 Barone, A., (2)449 Andersson, T., (2)449 Barros-Neto, J., 480, 519 Andrews, G. E., 570 Bartholdi, E., 681, (2)435, (2)449, (2)450 Angel, Y. C., (2)446, (2)450 Barton, G., (2)117, (2)449 Anger, C. T., 470, 473, (2)455 Bary, N. K., 78 Anischenko, S. E., (2)406, (2)449 Basinger, J. O., (2)41

824

Author index 825

Bass, M., (2)358, (2)361 Bodenhausen, G., (2)435, (2)406 Bassani, F., (2)1, (2)249, (2)406, (2)376, (2)396, Bogner, R. E., (2)178 (2)403, (2)404 Bogolyubov, N. N., (2)117 Basset, A. B., 468 Bohr, N., (2)430 Bastiaans, M. J., (2)406 Bokut, B. V., (2)349 Bedrosian, E., 184, 247, 403, 455, 458, 674, (2)43, Bolton, H. C., (2)248, (2)450 (2)131 Boltzmann, L., (2)63, (2)362 Bell, R. J., (2)303 Bona, J. L., 633 Bell, S. R., 141 Bonzanigo, F., 680 Beltrami, E. J., 479, 519, (2)113, (2)117 Booij, H. C., 8, (2)451 Beltzer, A. I., 8, (2)450 Boole, G., 234, 246 Belvaux, Y., 8 Borel, F. E. J. E., 55, 388, (2)64 Bendat, J. S., (2)116, (2)179 Born, M., (2)248, (2)348, (2)424, (2)425, (2)429, Benedek, A., (2)69 (2)432 Benedetto, J. J., 78, 79, 525 Bortz, M. L., (2)303 Benjamin, T. B., 624, 628, 629, 630, 635 Bose, N. K., 681, (2)1 Bennett, C., 248, 414, 415 Boswarva, I. M., (2)348 Bennett, C. L., (2)116 Bowlden, H. J., (2)303 Bennett, W. R., (2)41 Boyd, D. W., 247 Berkson, E., 414, (2)42 Boyd, R. W., (2)352, (2)356, (2)403, (2)404 Bernstein, S. N., 343, 344 Bracewell, R. N., 285 Berreman, D. W., (2)303 Brachman, M. K., 8, 285, (2)247, (2)450 Berthelot, Y. H., (2)451 Brackx, F., (2)41 Bertie, J. E., 712, 713, 732, (2)303 Braga, E. P., (2)450 Bertin, J. J., 585 Brandenburger, G. H., (2)406, (2)406 Bertrand, G., 57, 59, 61, 79, 107, 108, 112, 119, Brau, C. A., (2)403 122, 148, 192, 223, 225, 313, 529, 533, 553 Braun, S., (2)180 Besicovitch, A. S., 141, 414 Brauner, N., (2)450 Besikovitch, A., see Besicovitch, A. S. Brazil, T. J., (2)179 Bessel, F. W., 19, 187, 293, 436, 453, 455, 456, Bremermann, H. J., 79, 487, 519, (2)42 457, 459, 460, 462, 464, 465, 469, (2)455, Bremmer, H., 79 (2)456, (2)457 Brezinski, C., 79 Bessis, D., (2)450 Brillouin, L., (2)248, (2)249, (2)250 Bhatia, A. K., (2)450 Brockman, M. W., (2)349 Bhatta, D., 8 Bronzan, J. B., (2)450 Bialecki, B., 731 Brown, B. M., (2)116 Bierens de Haan, D., 5, 8 Brown, F. C., (2)348 Bierman, G. J., 632 Brown, Jr., J. L., 247, (2)179 Birkhoff, G., 77 Brown, J. W., 78 Birman, J. L., (2)249 Brychkov, Yu. A., 519 Bitsadze, A. V., (2)41 Buchkovska, A. L., (2)69 Blaauw, H. J., (2)406 Buck, J. R., 8, 680, 681 (2)406, (2)406 Black, E., 248 Buckingham, A. D., (2)348 Blackwell, C. A., (2)179 Budde, B. A., 303 Bland, S. R., 585 Bueckner, H. F., 632 Blankenbecler, R., (2)406 Buffoni, B., 414 Blaschke, W. J. E., (2)255, (2)256, (2)257, (2)303, Bülow, T., (2)179 (2)304, (2)412, (2)416 Bulirsch, R., 706 Blasco, O., (2)69 Buniakowski, V. Y., 58, 205, 207, 217, 220, 287, Bleistein, N., 434 343, 362, 387, (2)101, (2)301 Bloch, F., (2)273, (2)303 Bunyakovsky, V. Y., see Buniakowski, V.Y. Bloembergen, N., (2)403 Burge, R. E., (2)412, (2)449 Blue, J. L., 731 Burk, F., 79 Blum, K., (2)450 Burke, P. G., (2)450 Blyumin, S. L., 681 Burkhardt, H., (2)117 Boas, Jr, R. P., 78, 414, 436, (2)49, (2)50, (2)51, Burkholder, D. L., 386 (2)52, (2)53, (2)54, (2)55, (2)56, (2)57, Burris, F. E., 680 (2)68, (2)70, (2)72 Butcher, P. N., (2)356, (2)403 Boche, H., (2)179 Butzer, P. L., 8, 9, 141, 246, 248, 253, 285, 328, Bochner, S., 78, 247, 328, (2)83 414, 681 Bode, H. W., 6, 23, (2)249, (2)450 Byron, Jr., F. W., (2)450

826 Author index

Cable, J. R., (2)449 Chen, Q., (2)179, (2)180 Cain, G. D., 247, (2)178 Chen, Y.-M., 328, 414, 416 Calderón, A. P., 207, 246, 248, 320, 413, 681, Chernyak, V., (2)403 (2)1, (2)2, (2)3, (2)9, (2)21, (2)22, (2)41, Chiao, R. Y., (2)406 (2)42, (2)59 Chinsky, L., 8 Caldwell, D. J., (2)349 Chlipala, J. D., (2)303 Callis, P. R., 8 Chow, C.-Y., 578, 585 Calvo, M. L., (2)179 Christ, M., (2)41, (2)406, (2)69 Campbell, G. A., 78 Chung, J., 519 Campbell, J., (2)67 Churchill, R. V., 78 Campbell, L. L., 414 Cima, J. A., 141, 328 Campos, J., (2)406 Cížek,V.,ˇ 680 Candan, Ç., (2)163, (2)180 Clancey, K. F., 585 Capitani, G., 681, (2)406 Clarkson, P. A., 633 Capobianco, M. R., 632, 731 Clausen, T., (2)455 Carbery, A., (2)63, (2)69, (2)70 Clausius, R. J. E., (2)339 Carcaterra, A., (2)179 Clough, S. A., (2)248 Cardona, M., (2)248 Cody, W. J., 285 Carleman, T., 550, 606, 608, 632 Cogswell, C. J., 8 Carleson, L., 327, 328 Cohen, G., (2)69 Carlin, H. J., (2)116 Cohn, D. L., 79, 248 Caron, A., (2)303 Coifman, R. R., 395, 398, 415, (2)41 Carrier, G. F., 78, 585, 632 Colavita, E., (2)406 Carrington, A., (2)303, (2)433 Cole, B., 328 Carro, M. J., (2)61 Cole, K. S., (2)240, (2)435, (2)436, (2)450 Carruthers, N. B., 585 Cole, R. H., (2)240, (2)304, (2)435, (2)436, (2)450 Carson, J. R., 6, (2)247, (2)450 Collocott, S. J., 731, 732 Cartan, H., (2)42 Colwell, P., (2)303 Carton-Lebrun, C., 247, 519 Condon, E. U., (2)180, (2)330, (2)331, (2)338, Cartwright, M. L., 78, 211 (2)341, (2)347, (2)349 Case, K. M., 633 Constantin, P., 634 Caspers, W. J., (2)403 Conte, S. D., 167 Castaño González, O. D., (2)248 Cooke, J. C., 632 Castriota, L. J., (2)116 Cooke, R. L., 248 Castro, F., (2)248 Cooper, L. N., (2)407 Catalan, E. C., 338, (2)457 Coppens, P., (2)449 Cataliotti, F. S., (2)403 Córdoba, A., 415, 416, (2)63, (2)69 Cauchy, A.-L., 103, 141 Córdoba, D., 416 Cauer, W., (2)209, (2)249 Corinaldesi, E., (2)117 Cayley, A., 38 Cormack, A. M., 277, 286 Corrington, M. S., (2)450 Cervený,ˇ V., 8 Cossar, J., 247, 328 Cesari, L., 328 Cotlar, M., 374, 414, (2)41, (2)65 Cesàro, E., 296, 301 Cotter, D., (2)356, (2)403 Chambers, W. G., (2)303 Cotton, A., (2)306, (2)335 Champeney, D. C., 20, 78, 247 Cottrell, D. M., 8, (2)406 Champion, P. M., 8 Courant, R., 79 Chan, D. A. S., (2)403 Cowalenko, N., (2)406 Chandarana, S., (2)69 Craig, M., (2)179 Chandler, G. A., 633 Cramer, G., 72 Chandrasekharan, K., 78, (2)83 Criscuolo, G., 731, 732 Chang, S.-L., (2)449 Crowe, J. A., (2)406 Chang, Y.-C., 142, 415 Crystal, T. H., 285 Charles, S. W., (2)304 Cusmariu, A., (2)160, (2)161, (2)169, (2)170, Chatfield, C., (2)180 (2)180, (2)181 Chaudhry, M. A., 519, (2)36, (2)37, (2)38, (2)42 Chawla, M. M., 731 Damelin, S. B., 731 Chebyshev, P. L., 365, 367, 439, 452, 456, 567, Damera-Venkata, N., 681, (2)178 568, 569, 695, (2)457 Danloy, B., 731 Chen, C.-H., (2)180 Davidon, W. C., (2)116 Chen, K.-K., 328, 414, (2)45 Davier, M., (2)406 Chen, K. Y., (2)180 Davies, B., 8

Author index 827

Davies, K. T. R., 78, 79 Dunkl, C. F., 328 Davies, R. W., 78, 79 Duoandikoetxea, J., 415, (2)25, (2)32, (2)41, Davis, B., 337, 338, 414, 584 (2)42, (2)61, (2)69, (2)179 Davis, J. A., 8, (2)179 Duren, P. L., 78, 337 Davis, K. M., 142, 415, 587 Dutta Roy, S. C., 681, (2)178 Davis, P. J., 730 Dykema, K. J., 695 Davydov, A. S., (2)248 Dyn’kin, E. M., 414, 415, (2)69 Dawson, H. G., 257 Dzhuraev, A., 632 de Dios Leyva, M., (2)248 De Alfaro, V., (2)249 Eastham, M. S. P., 328 De Bonis, M. C., 731 Echeverría, J. C., (2)180 De Carli, L., 247 Edge, C. M., (2)406 De Heer, F. J., (2)449, (2)450 Edmunds, D. E., 416 De Knock, B., (2)41 Edwards, R. E., 78 De Schepper, H., (2)41 Eelbode, D., (2)41 De Vries, G., 624, (2)349 Ehlers, J. F., 7 Deans, S. R., 286 Ehrenreich, H., (2)303 Debiais, G., (2)248 Einstein, A., (2)279 Debnath, L., 8, 633, 681 Eiselt, M., 681 Debye, P., (2)293 Eklund, H., (2)449 Degasperis, A., 633 Eleftheriades, G. V., (2)406 Del Pace, C., 585 Elliott, D., 158, 226, 230, 585, 632, 730, 732, Delanghe, R., (2)41 (2)42, (2)43 Deléchelle, E., (2)180 Emeis, C. A., (2)349 Ephremidze, L., 248, (2)69 Della Vecchia, B., 731 Erdélyi, A., 8, 256, 259, 267, 276, 285, 453, 456, Delves, L. M., 730 468, 471, 521, 522, 568, 569 Derriennic, Y., (2)69 Ernst, R. R., 681, (2)434, (2)435, (2)449, (2)450 Desoer, C. A., (2)116 Ervin, V. J., 633 Dexter, D. L., (2)242, (2)406, (2)406, (2)406, Essén, M., 328, 333, 411, 412, 413, 414 (2)406, (2)303, (2)406 Essex, J. W., (2)406 Diaz-Santana, L., (2)179 Estrada, R., 585, 632, 633 Dickinson, B. W., (2)180 Eu, J. K. T., (2)178 Dienstfrey, A., (2)248 Euler, L., 74, 522, (2)340, (2)454, (2)457 Diethelm, K., 731 Evans, B. L., 681, (2)406 Dignam, M. J., (2)304 Eyre, D. J., (2)248 Ding, J.-J., (2)180 Eyring, H., (2)349 Dirac, P. A. M., 41, 63, 65, 69, 79, 99, 222, 464, Eysel, H. H., 732 474, 483, 485, 488, 492, 512, 513, 514, 682, (2)56, (2)111, (2)114, (2)457 Fabes, E. B., (2)41 Dirichlet, J. P. G. L., 17, 296, 298, 299 Falb, P. L., (2)116 Divin, Yu. Ya., (2)406, (2)449 Fan, K. L., (2)406, (2)406 (2)406 Doelitzsch, D. F., (2)136 Fang, P. H., (2)450 Dolgov, O. V., (2)304 Fannin, P. C., (2)304 Domínguez, M., 416 Fano, U., (2)303 Donovan, A. F., 699, 731 Faraday, M., (2)306, (2)307, (2)309, (2)319 Dorf, R. C., (2)178 Farmer, C. M., (2)450 Dow, M. L., 632 Farrell, P., 732 Dows, D. A., (2)303 Farshait, P. G., 585 Drachman, R. J., (2)450 Fatou, P., 94, 117, 320, 326, 328, 334, 353 Drago, R. S., (2)450 Fefferman, C., 395, 398, 415, (2)10, (2)41, (2)42 Dragomir, N. M. 732 Fefferman, R., 415 Dragomir, S. S., 732 Feinberg, G., 8 Drude, P., (2)223 Fejér, L., 296, 299 Drazin, P. G., 633 Fekete, P. W., 8 Duan, J., 633 Feldman, M., (2)179, (2)180 Dubau, J., 8 Felland, M., 633 Duesler, E., (2)303 Felsberg, M., (2)179 Duffin, R. J., 8, 675, 676, 677, 678, 681, (2)41 Feng, B.-F., 633 Duggal, B. P., 246 Feng, Z. W., (2)449 Dugundji, J., (2)179 Fernández-Cabrera, L. M., 587 Dumbrajs, O., (2)450 Fernández-Castaño, C., 416, (2)66, (2)449

828 Author index

Ferrand, J., 681 Garsia, A. M., 248, 382, 415, (2)65, (2)69 Ferrando, S., (2)41 Gasquet, C., 495, 519 Ferrell, R. A., (2)303 Gauss, J. C. F., 472, 694, 695, 696, 728, (2)455 Ferro Fontán, C., (2)250 Gautschi, W., 699, 731, 732 Ferry, J. D., 8, (2)451 Gazonas, G. A., 731 Feshbach, H., 78 Gegenbauer, L. B., 439, 452, 472, 570, (2)455 Fessler, T., 79 Geitz, M., (2)406 Fiddy, M. A., (2)406, (2)449 Gel’fand, I. M., 79, 516, 517, 519 Fink, A. M., 80 Gell-Mann, M., (2)248 Fischer, J., (2)250, (2)450 Gentile, S., (2)406 Flannery, B. P., 285, 680, 699, 711, 712, 713, 730, Gera, A. E., 632 (2)406 Gerjuoy, E., (2)428, (2)429, (2)432, (2)450 Flett, T. M., 328, 363, 414, (2)45 Giacovazzo, C., (2)449 Fogale, F., 681 (2)406 Giambiagi, J. J., (2)116 Fokas, A. S., 633 Giancaspro, C., (2)449 Folland, G. B., 79, 415 Gillespie, T. A., 414, (2)42 Fontana, L., 248 Gindikin, S., 286 Fontelos, M. A., 416 Ginzberg, V. L., see Ginzburg, V. L. Ford, W. F., 79 Ginzburg, V. L., (2)248, (2)303, (2)446, (2)447 Forelli, F., 413, 415 Giova, R., 731, 732 Foreman, A. J., 633 Girard, A., 235, 248 Fort, C., (2)406 Giraud, G., (2)41 Foster, R. M., 78 Glaeske, H.-J., 247 Foucher, J., (2)449 Glasser, M. L., 79, 150, 152, 155, 246 Fourès, Y., (2)116 Gledhill, R. J., (2)406 Fourier, J. B. J., 252, (2)12 Gloersen, P., (2)406, (2)406, (2)406 Franken, P. A., (2)357, (2)406, (2)406 Glover, III, R. E., (2)303 Fratila, R., (2)116 Goedecke, G. H., (2)303 Fredholm, E. I., 2, 589, 592, 594 Gohberg, I., 247, 632 Freedman, M. I., (2)116 Gokhberg, I. Ts., 247 French, R. H., (2)303 Golberg, M. A., 585, 632 Fresnel, A. J., (2)454, (2)456 Gold, B., 680, 681 Fried, B. D., 167 Goldberg, R. R., (2)69 Fritz, T., (2)248 Fröhlich, H., (2)247 Goldberger, M. L., (2)116, (2)117, (2)239, (2)248, (2)449 Fromme, J. A., 585, 632 Frye, G., (2)219 Good, Jr., R. H., (2)403 Fubini, G., 21, 55, 56, 60, 119, 268, 357, 367, 377, Goodspeed, F. M., 632 504, 506, 613, (2)78 Gopinath, D. V., 732 Fubini, S., (2)406 Gori, L., 732 Fuchssteiner, B., 633 Gornov, E., (2)303 Fukao, Y., 584 Gorter, C. J., (2)240, (2)247, (2)303 Fuks, B. A., (2)42 Gottlieb, M., (2)303 Fung, Y. C., (2)406 Gottlieb, O., (2)180 Funk, P., 277 Goursat, E. J.-B., 29, 90 Furlan, G., (2)406 Goutte, R., (2)1 Furuya, K., (2)303, (2)348 Gradshteyn, I. S., 522 Futterman, W. I., (2)248 Grafakos, L., 247, 662, 664, (2)69 Graham, G., (2)249 Gabisonija, I., 681 Graham, I. G., 633 Gabor, D., 446, (2)119, (2)127 Gram, J. P., 696 Gakhov, F. D., 8, 78, 79, 112, 544, 584, 585, 632 Gray, A., 465, 466, 467 Gallardo, D., (2)69 Green, G., 331, 333, 341, (2)425 Galusek, D., (2)303 Green, M., 179 Gamelin, T. W., 328 Greenaway, A. H., (2)406, (2)449 Gao, M.-Z., 681 Greenaway, D. L., (2)303 Gaposhkin, V. F., 414 Greene, R. E., 78 García, A. G., (2)180 Greengard, L., (2)248 Garcia-Cuerva, J., 415, (2)61 Gregor, J., 732 Gårding, L., 211, 247 Grennberg, A., (2)179 Garnett, J. B., 78, 248, 386, 415, 416, (2)69 Griessbach, G., 681

Author index 829

Gross, B., 731, (2)116, (2)247, (2)248, (2)304, Hermite, C., 439, 446, 447, 448, 450, 695, (2)119, (2)450, (2)451 (2)154, (2)155, (2)156, (2)157, (2)163, Grosse, P., (2)303 (2)165, (2)455 Gründler, R., (2)249 Herring, F. G., (2)450 Grushevskii, S. P., 414 Hewitt, E., 211, 328 Gubankov, V. N., (2)406 Heywood, P., 246, (2)69, (2)70 Guillemin, E. A., (2)116, (2)450 Higgins, J. R., 196, 247, 414 Gundy, R. F., 386 Hilb, E., 141 Gurielashvili, R. I., 414 Hilbert, D., 3, 84, 117, 132, 328, 632 Güttinger, W., 519 Hilgevoord, J., (2)116, (2)117 Hill, A. E., (2)406, (2)406 Hadjichristov, G. B., (2)116, (2)403 Hille, E., 211, 215, 247 Haffad, A., (2)450 Hinich, M. J., (2)179 Hagan, D. J., (2)406 Hinojosa, J. H., 8 Hahn, S. L., 8, 84, 247, 471, 680, (2)178, (2)179 Hirschman, Jr., I. I., 359 Hämäläinen, R. M. K., (2)248 Hobson, E. W., 141, 247, 328 Hamel, G., 584 Hochstadt, H., 8, 78, 537, 632 Hamilton, J., (2)117, (2)249, (2)450 Hoenders, B. J., (2)417, (2)449 Handelsman, R. A., 434 Hölder, O., 12, 13, 45, 54, 58, 59, 60, 61, 79, 80, Hanke, K., (2)406 81, 96, 98, 107, 111, 188, 189, 206, 209, 210, Hankel, H., 1, 453, 464, 465 224, 230, 231, 356, 362, 389, 390, 391, 392, Hans, V., (2)179 393, 394, 396, 408, 532, 537, 685, (2)4, (2)5 Hänsch, T. W., (2)406 Holbrow, C. H., (2)116 Hansen, E., 178, 179, 246, 347, 525 Holland, A. S. B., 78 Harbecke, B., 732 Hollenbeck, B., 247, 328 Harbeke, G., (2)303 Hopfield, J. J., (2)249 Hardy, G. H., 3, 5, 6, 8, 43, 57, 59, 61, 79, 85, 96, Hörmander, L., 247 99, 107, 108, 112, 117, 118, 119, 120, 121, Horton, Sr., C. W., (2)450 122, 132, 141, 148, 150, 154, 170, 192, 204, Horváth, J., 519, (2)33, (2)34, (2)35, (2)41, (2)42, 211, 215, 223, 225, 246, 313, 328, 336, 354, (2)60 355, 357, 358, 359, 363, 368, 372, 373, 374, Houghton, E. L., 585 379, 389, 414, 415, 527, 529, 533, 553, 661, Hounsfield, G. N., 277 681, (2)45, (2)46 Howard, R. E., (2)348 Harte, W. E., (2)349 Howell, K. B., 21, 78, 519, 680, (2)124 Hartley, R. V. L., 1, 262, 263, 264, 285, (2)410, Huang, J. B., (2)303 (2)454 Huang, N. E., (2)170, (2)171, (2)173, (2)174, Hartogs, F., (2)39 (2)178, (2)180 Hasegawa, T., 731, 732, (2)406, (2)406 Huang, W., (2)180 Hassani, S., 78 Hughes, E., 519 Hauss, M., 328 Hughes, M. S., (2)406 (2)406, (2)406 Hayashi, N., 633 Hulthén. R., (2)248 Hunt, R. A., 328, 354, 388, 395, 397, 399, 403, Hayes-Gill, B. R., (2)406 415, 679, 680, 683 He, P., (2)446, (2)450 Hunter, D. B., 730, 732 Healy, W. P., (2)343, (2)349 Huntington, H. B., 633 Heaviside, O., 450, 485, 490, (2)114, (2)119, Hutchings, D. C., (2)403 (2)139, (2)141, (2)143, (2)145, (2)148, Hutchinson, D. A. (2)349 (2)455 Hutchinson, J. S., 8 Heinig, H. P., (2)57 Hutt, P. K., (2)450 Heisenberg, W. K., 475, 486, (2)115, (2)118, (2)244, (2)431 Ichimura, M., (2)249 Helgason, S., 286 Iftekharuddin, K. M., 8 Helleloid, G. T., 717 Igari, S., 328 Hellinger, E., 141 Inagaki, T., (2)267, (2)303 Helmholtz, H. L. F. von, (2)201, (2)276 Inguscio, M., (2)406 Helson, H., 328, 386, 388, 414, (2)42 Ino, Y., (2)406 Henery, R. J., 731 Inokuti, M., (2)248, (2)406 Henrici, P., 59, 78, 79, 83, 112, 141, 731 Ishida, H., 731, (2)303 Henry, C. H., (2)348 Ishikawa, S., 519 Herdman, T. L., 8 Islam, M. M., (2)406 Herglotz, G., (2)208, (2)209, (2)216, (2)249 Ivanov, I. A., 8 Herman, G. T., 8 Ivi´c,A., 665

830 Author index

Iwaniec, T., (2)41, (2)42 Kellogg, O., 84, 122, 132 Iwasa, Y., (2)406, (2)406 Kelvin (Lord), (2)362 Kendig, M. W., (2)450 Jackson, J. D., (2)93, (2)239, (2)247, (2)303, Kenig, C. E., 247, 633 (2)304, (2)348 Kerr, F. H., (2)180 Jackson, L. B., (2)178 Kerr, J., (2)306 Jacobi, C. G. J., 439, 452, 472, 695, (2)456 Kestelman, H., 57, 79 Jahoda, F. C., (2)303 Ketolainen, P., 732 Jajte, R., (2)69 Khabayev, P. V., (2)406, (2)449 Jaswon, M. A., 633 Khuri, N. N., (2)449 Jaw, S.-B., 681, (2)116, (2)179 Khvedelidze, B. V., 585 Jaynes, E. T., (2)450 Kierat, W., 519 Jeffrey, A., 78 Kikuchi, M., 584 Jensen, J. L. W. V., (2)97, (2)99, (2)100 Kilbas, A. A., 285 Jensen, L. H., (2)421 Kim, W.-J., (2)69 Jiang, H.-M., 416 Kim, Y. S., (2)450 Jiang, S., 632 Kimel, I., (2)250, (2)303 Jichang, K., 681 King, F. W., 79, 564, 695, 717, 731, 732, (2)238, Jin, Y. S., (2)117, (2)249 (2)240, (2)241, (2)243, (2)248, (2)249, Joachain, C. J., (2)450 (2)250, (2)267, (2)303, (2)304, (2)348, Jodeit, Jr., M., 247 (2)349 Johansson, J., (2)449 Kinukawa, M., 328 Johnson, D. W., 732, (2)304 Kircheva, P. P., (2)116, (2)403 Johnson, R. S., 633 Kirzhnits, D. A., (2)304 Jolly, L. B. W., 291, 292, 525, 526 Kishida, H., (2)385, (2)406 Jones, D. S., 518, 519 Kittel, C., (2)303, (2)407 Jones, P. W., 415, (2)59, (2)304 Klein, A., (2)449 Jones, R. L., 246, (2)406 Klein, N., (2)406 Jones, W., (2)303 Klucker, R., (2)303 Joo, T., (2)449 Kneizys, F. X., (2)248 Jordan, M. E. C., 30, 31, 101, 239, 425, 433, 713, Knight, B. W., 731, (2)248, (2)304 (2)86 Kober, H., 38, 79, 141, 170, 204, 212, 215, 246, Josephson, B. D., (2)406, (2)407, (2)409, (2)410 247, 408, (2)47, (2)48, (2)68, (2)69 Journé, J.-L., (2)41 Kochneff, E., 202, 246, 346, (2)42 Kaczmarz, S., 141 Koda, T., (2)406, (2)406 Kador, L., (2)404 Köhler, P., 731 Kaestner, A., (2)406 Kogan, Sh. M., (2)403 Kahanpää, L., 415 Kohlenberg, A., 414, (2)449 Kak, S. C., 285, 680 Kohlmann, K., 8 Kalisch, H., 633 Koizumi, S., 413, 681, (2)69 Kalton, N. J., 247 Kokilashvili, V. M., 415, 416 Kamychine, V., (2)406 Koláˇr,P., (2)450 Kanazawa, A., (2)117 Kolm, P., 730 Kane, G. L., (2)450 Kolmogoroff, A., see Kolmogorov, A. Kaneko, M., 414, 415 Kolmogorov, A., 335, 336, 337, 338, 362, 364, Kanter, H., (2)409 367, 408, 414, 416 Kantorovich, L. V., 78, (2)42 Komori, Y., 681 Kanwal, R. P., 79, 492, 519, 585, 632, 633, (2)33 Koosis, P., 78, 141, 142, 328, 338, 369, 386, 414, Karl, J. H., 8 415, (2)69 Karstens, W., (2)248 Koppelman, W., 585 Karttunen, K., 732 Korsunsky, S. V., (2)406, (2)406 Katz, N. H., 416 Korteweg, D. J., 624 Katznelson, Y., 78 Kosarev, E. L., (2)409, (2)449 Kaufmann, B., (2)449 Koshelets, V. P., (2)406, (2)406 Kaup, D. J., 633 Kosloff, D. D., 731 Kawahara, T., 633 Kotel’nikov, V. A., 344, (2)419, (2)420 Kawasaki, T., (2)403 Kotelyanskii, I. M., (2)406 Kawata, T., 247, (2)267 Kozima, K., 303 Keast, P., 731 Krajca, V., 681 Keˇcki´c,J. D., 78, 585 Krall, N. A., (2)428, (2)432, (2)450 Keefe, C. D., (2)247 Kramer, H. P., (2)449

Author index 831

Kramers, H. A., 6, 23, 435, 708, (2)182, (2)184, Larkin, K. G., 8 (2)187, (2)189, (2)190, (2)195, (2)197, Larkin, S. Y., (2)406, (2)449 (2)199, (2)201, (2)206, (2)207, (2)208, Larmor, J., (2)273 (2)209, (2)212, (2)213, (2)215, (2)216, Larsen, R., 246 (2)217, (2)244, (2)247, (2)250, (2)273, Lasser, R., 328 (2)290, (2)298, (2)305, (2)376, (2)377, Laurent, P. A., 25, 31, 32, 33, 646, (2)291, (2)294 (2)378, (2)385, (2)386, (2)388, (2)390, Lauwerier, H. A., 498, 519 (2)391, (2)392, (2)398, (2)402, (2)444 Lax, P. D., 634 Krantz, S. G., 38, 78, 328, (2)39, (2)42 Le Van Quyen, M., (2)449 Krbec, M., 415 Lebedev, V. I., 732 Krein, M., (2)249 Lebesgue, H.-L., 21, 45, 49, 51, 52, 53, 54, 55, 61, Kress, R., 8, 585, 632, 732 63, 66, 126, 128, 177, 217, 228, 229, 241, Kröger, K., 303 244, 254, 259, 287, 327, 353, 368, 711, Kronecker, L., 18, 298, 442, 639, (2)280 (2)193, (2)199, (2)299 Kronig, R. De L., 6, 23, 435, 708, (2)116, (2)182, Lee, C. C., (2)450 (2)184, (2)187, (2)189, (2)190, (2)195, Lee, C. M., (2)450 (2)197, (2)199, (2)201, (2)205, (2)207, Lee, M. H., 471, (2)248, (2)303 (2)208, (2)209, (2)212, (2)213, (2)215, Lee, S.-Y., 8, 116, (2)449 (2)216, (2)217, (2)225, (2)240, (2)242, Lee, Y. W., 6 106, (2)450 (2)247, (2)250, (2)273, (2)290, (2)298, Legendre, A.-M., 438, 439, 441, 442, 443, 444, (2)303, (2)304, (2)305, (2)376, (2)377, 452, 563, 564, 565, 567, 694, 695, 697, 700, (2)378, (2)385, (2)386, (2)388, (2)390, 728, (2)456 (2)391, (2)392, (2)398, (2)402, (2)444 Lei, Y., (2)406 Krook, M., 78, 585, 632 Leibnitz, G. W., 164 Krupnik, N. Ya., 247, 328, 632 Lekishvili, M. M., 328 Kruskal, M. D., 625 Lemoine, J., (2)180 Krylov, V. I., 704 Leontovich, M., (2)304 Kubo, R., (2)249, (2)403 Lerch, M., (2)457 Kuchiev, M. Yu., (2)450 Lerche, I., 633 Kuethe, A. M., 578, 585 Lerner, R. M., (2)179 Kuhn, W., (2)225, (2)284, (2)324, (2)340 Leuthold, P. E., (2)179 Kuijlaars, A. B. J., 414 Lévêque, G., (2)249, (2)303 Kuipers, L., 585 Levi-Civita, T., (2)337 Kumar, S., 731 Levin, B. Ja., (2)417, (2)449 Kumaresan, R., (2)179 Levin, D., 74, 76, 77, 82, 165, 718 Kummer, E. E., 257, (2)455 Levine, R. D., (2)89, (2)116 Kunz, K. S., (2)247 Levinson, N., 141, 247, 585 Kuramoto, Y., 633 Lewin, L., 632 Kurths, J., (2)449 L’Huillier, A., (2)358 Kutay, M. A., (2)163, (2)406 Li, C.-P., (2)180 Kutta, M. W., 580 Li, L., (2)179 Kuwata, H., (2)406, (2)303 Li, X., 633, (2)69 Kuwata-Gonokami, M., (2)406 Liang, Z., (2)449 Kuzmenko, A. B., (2)247 Lichvár, P., (2)303 Lacey, M. T., (2)59, (2)69 Lidiard, A. B., (2)348 Lachaux, J.-P., (2)449 Lifanov, I. K., 632 Ladopoulos, E. G., 633 Lifshitz, E. M., (2)240, (2)247, (2)249, (2)304, Laeng, E., 247, 681 (2)449 Laguerre, E. N., 439, 449, 694, 695, (2)455 Lighthill, M. J., 79, 509, 518 Lahham, M., (2)450 Likharev, K. K., (2)449 Laigle, A., 8 Lin, M., (2)69 Lamb, Jr., G. L., (2)248 Lindelöf, E. L., 100, 201, (2)194, (2)197, (2)204, Lamperti, J., 328 (2)244, (2)278 Lan, Z., (2)303 Lindsey, III, J. F., (2)136 Landau, E., 78 Liouville, J., 34, 70, (2)364 Landau, L. D., (2)240, (2)247, (2)249, (2)304, Lipschitz, R. O. S., 12, 13, 98, 316, 317, (2)3, (2)5 (2)449 Liška, M., (2)303 Langley, R. S., (2)179 Littlewood, J. E., 80, 99, 117, 132, 141, 246, 328, Lapitskaya, I. L., (2)406, (2)406 336, 354, 355, 357, 358, 359, 363, 368, 372, Laplace, P.-S., 1, 267, 268, 269 373, 374, 379, 389, 414, 415, 661, 681, Laramore, G., (2)348 (2)45, (2)406

832 Author index

Liu, C. H., (2)406 Martini, A. F., (2)450 Liu, H. H., (2)406, (2)406, (2)406, (2)406 Martinerie, J., (2)406 Liu, H.-P., 731 Martinis, M., (2)450 Liu, Y.-C., (2)236, (2)251, (2)304 Martín-Reyes, F. J., (2)66, (2)69 Löfström, J., 416 Mashreghi, J., (2)69 Logan, B. F., 350, 414, 585, (2)179 Mastroianni, G., 585, 731, 732 Lohmann, A. W., (2)147, (2)148, (2)149, (2)178, Matheson, A. L., 141, 328 (2)179 Mathews, G. B., 465, 466, 467 Long, S. R., (2)406, (2)406, (2)406, (2)406,(2)180 Matsuno, Y., 246, 621, 624, 633 Longman, I. M., 731 Matsuoka, K., (2)69 Loomis, L. H., 234, 236, 237, 239, 248, 382, 385, Maximon, L. C., (2)250 414 Maxwell, J. C., (2)201, (2)216, (2)275, (2)280, Lorentz, H. A., (2)93, (2)186, (2)223, (2)239, (2)332 (2)244, (2)247, (2)313, (2)314, (2)324, McBride, A. C., (2)180 (2)330, (2)339, (2)359 McCabe, J. H., 285 Lorenz, L., (2)339 McCaslin, S. R., 681, (2)406 Loughlin, P. J., (2)179 McClellan, J. H., (2)180 Love, E. R., 79, 192, 247 McClure, J. P., 422, 434 Lovell, R., (2)248 McCormick, B. W., 585 Lowenthal, S., 8 MacDonald, J. R., 8, 285, (2)247, (2)248 Lubinsky, D. S., 730, 731 McDowell, M. R. C., (2)450 Lucarini, V., (2)248, (2)249, (2)406, (2)303, McGroddy, J. C., (2)349 (2)406, (2)376, (2)385, (2)386, (2)403, McLachlan, A. D., (2)303, (2)433 (2)404 Maclaurin, C., 684, 688 Ludwig, F., (2)449 McLean, W., 158, 226, 230, (2)42, (2)43 Lukashenko, T. P., 141 McNamara, D. E., 8, (2)406 Lund, A. D., 695 McQuarrie, D. A., 78 Lusin, N. N., 141, 325, 326, 327 MacRobert, T. M., 465, 466, 467 Lutz, A., (2)406 Medvedev, B. V., (2)117 Lyness, J. N., 8, 731 Mehler, F. G., 460, 465, (2)156 Lyubarskii, Y. I., 681 Meˇıman,N. N., (2)248 Mejlbro, L., 415 Madych, W. R., 8 Mellin, R. H., 1, 269, 270, 271, 273, 275 Maeda, S., (2)303 Melrose, D. B., (2)304 Magnus, W., 8, 256, 259, 267, 276, 285, 453, 456, Mendlovic, D., (2)147, (2)148, (2)406, (2)406 468, 471, 521, 522, 568, 569 Menon, M. J., (2)450 Mahler, G., (2)249 Menzel, J., (2)406 Majda, A., 634 Merzbacher, E., (2)108, (2)116 Maker, P. D., (2)406 Meskhi, A., 416, 681 Maksimov, E. G., (2)304 Meyer, R. M., 78 Malloy, K. J., (2)406 Meyer, Y., 415, (2)41 Malmstén, C. J., 468 Mezincescu, G. A., (2)304 Mandal, B. N., 632 Michor, P., 286 Mandelstam, S., (2)1 Mickus, K. L., 8 Mangulis, V., 461, (2)450 Mihlin, S. G., see Mikhlin, S. G. Manogue, C. A., (2)260, (2)267, (2)303 Mikhlin, S. G., 8, 632, (2)41 Mansfeld, F., (2)450 Millane, R. P., (2)116 Mantese, J. V., (2)248 Miller, D. A. B., (2)404 Maragos, P., (2)179 Miller, J. G., (2)406, (2)450, (2)451 Marathay, A. S., (2)449 Miller, K. S., 38 March, N. H., (2)303 Miller, S. M., (2)179 Marcinkiewicz, J., 141, 365, 372, 373, 404, 413 Miller, W. H., (2)417 Marichev, O. I., 285 Milton, G. W., (2)248 Markushevich, A. I., 78 Minerbo, G., (2)116 Marsden, M. J., 681 Minkowski, H., 205, 207, 230, 231, 334, 337, 354, Marshall, A. G., (2)410, (2)449, (2)450 359, 362, 374, (2)23, (2)30, (2)46 Martensen, E., 732 Misell, D. L., (2)449 Martin, A., (2)117, (2)249 Mishnev, A. F., (2)420, (2)449 Martin, B. G., (2)450 Mitrinovi´c,D. S., 78, 80, 585 Martin, G., (2)42 Mittag–Leffler, M. G., 35 Martin, P. C., (2)303, (2)304 Mobley, J., (2)406, (2)450, (2)451

Author index 833

Moezzi, H., (2)449 Nho, W., (2)179 Moffitt, W., (2)349 Niang, O., (2)180 Mohankumar, N., 732 Nicholson, J. W., 465 Mojahedi, M., (2)249 Nickel, K., 632 Molina, A., (2)304 Nielsen, U., (2)303 Monegato, G., 585, 732 Nieto-Vesperinas, M., (2)1, (2)449 Montgomery, H. L., 681 Nikulin, A. Yu., 449 Mook, D. T., (2)116 Nishijima, K., (2)117, (2)450 Moore, D. W., 633 Nitsche, R., (2)248 Morán, M. D., (2)70 Noda, I., (2)179 Morawitz, H., 730 Noether, F., 632 Morera, G., 30 Nowak, M. D., (2)179 Morlet, A. C., 633 Nozières, P., (2)248, (2)280, (2)281, (2)282, Morse, P. M., 78 (2)285, (2)303, (2)304, (2)337 Moscowitz, A., (2)349 Nussenzveig, H. M., 519, 731, (2)1, (2)105, Moss, T. S., (2)242 (2)116, (2)117, (2)208, (2)242, (2)248, Moss, W. F., 585 (2)249, (2)406, (2)406, (2)304, (2)335, Mossotti, O. F., (2)339 (2)342, (2)349, (2)449 Motter, A. E., (2)450 Nuttall, A. H., 247 Mouton, H., (2)306 Msezane, A. Z., (2)450 Oberhettinger, F., 8, 256, 259, 267, 276, 285, 453, Muckenhoupt, B., 354, 388, 395, 397, 399, 403, 456, 468, 471, 521, 522, 568, 569 405, 407, 414, 415, 416, 679, 680, 683 Occorsio, D., 585, 731 Mueller, F. M., 732 Occorsio, M. R., 732 Mukamel, S., (2)403 O’Connell, J. S., (2)250 Oden, J. T., 78 Mukhtarov, Ch. K., (2)247, (2)348 O’Donnell, M., (2)450 Murakami, T., (2)450 Offermann, V., (2)303 Murthy, A. S. V., 633 Ohmura, T., (2)116 Musienko, T., (2)304 Ohta, K., 731 Muskhelishvili, N. I., 8, 59, 61, 78, 79, 141, 313, Ojeda-Castañeda, J., (2)179 549, 585, 632 Okada, S., 247, 584, 585 Okikiolu, G. O., (2)67, (2)68, (2)69, (2)72 Nabarro, F. R. N., 619, 633 Okubo, S., (2)236, (2)251, (2)304 Nabet, B., (2)248 Olejniczak, K. J., 285 Nabighian, M. N., 8 Oleszkiewicz, K., 681 Nadir, M., 732 Olhede, S., (2)180 Naegele, K., (2)303 Olkkonen, H., (2)178 Nagel, A., (2)62, (2)63, (2)69 Olmstead, W. E., 632 Nahin, P. J., (2)179 Olver, F. W. J., 426, 434 Nakajima, N., (2)449 O’Neil, R., 247 Nakano, J., (2)179 Onishchuk, O. V., 585 Namias, V., (2)180 Ono, H., 624, 626, 628, 629, 630, 635 Nash, P. L., (2)303 Onsager, L., (2)289 Natarajan, A., 732 Oosterhoff, L. J., (2)349 Naumkin, P. I., 633 Oppenheim, A. V., 8, 680, 681, (2)116, (2)126, Nawab, S. H., (2)116 (2)178 Nayak, A. R., 732 Ortner, N., (2)42 Nayfeh, A. H., (2)116 Orton, M., 498, 519, 633 Nazarov, F., 415 Osgood, W. F., (2)42 Needham, T., 78 Osilenker, B. P., 414, 415 Negrescu, C., 681 Oskolkov, K., 681 Nehari, Z., 38 Oswald, J. R. V., (2)179 Neri, U., 253, (2)3, (2)24, (2)25, (2)41, (2)42 Ozaktas, H. M., (2)163, (2)180 Nessel, R. J., 9, 141, 246, 248, 253, 285, 328 Neufeld, J. D., (2)303 Paciorek, K. A., 285 Neuman, C. P., 730 Padala, S. K., 681 Neumann, C. G., 599 Paes, J. T. S., (2)406 Neumann, F., 564 Page, C. H., (2)116, (2)249, (2)450 Nevai, P., 171, 246 Paget, D. F., 730, 732 Nevanlinna, R., 100, (2)249 Päivärinta, L., 416 Newell, A. C., 79 Pake, G. E., (2)450

834 Author index

Pal’cev, A. A., 704 Phragmén, E., 100, 201, (2)194, (2)197, (2)204, Paley, R. E. A. C., 141, 215, (2)95, (2)98, (2)116 (2)244, (2)278 Palmer, K. F., 303 Pichorides, S. K., 203, 247, 318, 328, 337, 343, Pan, S., (2)406 361, 413, 414, 417 Pan, X., 732 Pick, L., 415 Pandey, J. N., 9, 59, 84, 132, 141, 196, 248, 328, Piersol, A. G., (2)179 490, 516, 517, 518, 519, 633, (2)35, (2)36, Piessens, R., 730, 732 (2)37, (2)38, (2)41, (2)42 Pikovsky, A. S., (2)449 Panman, P., (2)69 Pilipovi´c,S., 519, (2)69 Panzone, R., (2)69 Pimentel, B. M., (2)450 Pao, Y.-H., 8, (2)249, (2)450 Pines, D., (2)248, (2)280, (2)281, (2)282, (2)285, Papadopoulos, S. F., 328 (2)303, (2)304, (2)337 Papoulis, A., 6, 20, 107, 141, 680, (2)180 Pinsky, M., 414 Parke, S., (2)248 Pinus, J. D., 585 Parker, P. J., (2)179 Pipkin, A. C., 8, 537, 632 Parks, P. C., (2)286 Pippard, A. B., (2)116 Parks, T. W., (2)180 Pišút, J., (2)406 Parris, D., (2)349 Pitaevskiˇı,L. P., (2)304 Parseval, M.-A., 21, 80, 170, 214, 217, 219, 221, Pitt, H. R., 212 254, 305, 329, 487, 490, 506, 517, 559, 562, Plancherel, M., 21, 80, 170 629, 641, 671, 673, (2)16, (2)28, (2)36, Planck, M. K. E. L., (2)105, (2)107, (2)432 (2)51, (2)120, (2)158, (2)232, (2)264, (2)271 Plaskett, J. S., (2)303 Pârvu, V., 681 Plemelj, J., 111, 112, 113, 295, 296, 545, 606, Patakova, I., 681 (2)41, (2)413 Patapoff, T. W., 8 Plessner, A., 117 Paternò, G., (2)449 Plieth, W. J., (2)303 Pathak, R. S., 519 Pochhammer, L. A., 257, 428, (2)454 Paulikas, A. P., (2)303 Podolsky, B., (2)247 Paveri-Fontana, S. L., 632 Poincaré, H., 57, 59, 61, 79, 107, 108, 112, 119, Pavlovskii, V. V., (2)406 122, 148, 192, 223, 225, 313, 529, 533, 553 Pearson, C. E., 78, 585, 632 Poisson, S.-D., 85, 86, 87, 88, 89, 90, 91, 92, 94, Peˇcari´c,J. E., 80 142, 222, 296, 339, 374, 375, 379, 380, 413, Pei, S.-C., 681, (2)116, (2)163, (2)168, (2)179, 468, 618, (2)48 (2)180 Polivanov, M. K., (2)117 Peierls, R., 618, 619, 621 Pollard, S., 141 Peiponen, K.-E., 732, (2)1, (2)248, (2)250, (2)303, Polyakov, P. L., 582 (2)385, (2)386, (2)396, (2)397, (2)402, Pólya, G., 80, 355, 661, 681,(2)46 (2)403, (2)404, (2)450 Polyanski, O. Yu., see Polyanskii, O. Yu. Pełczynski, A., 247 Polyanskii, O. Yu., (2)406, (2)406 Pelzl, P. J., 79, 717 Ponce, G., 633 Penyaz, V. A., (2)349 Popov, G. Ya., 585 Pereyra, C., 416 Poppe, U., (2)406 Pérez Alvarez, R., (2)248 Porras, I., 79 Peˇrina,J., (2)413, (2)449 Portis, A. M., (2)403 Perry, P. A., (2)179 Potamianos, A., (2)179 Petermichl, S., 416 Poularikas, A. D., 8, 471, 680 Peters, A. S., 536, 537, 584, 609 Power, E. A., (2)349 Peters, C. W., (2)406, (2)406 Prabhakar, J. C., 8 Peters, J. M. H., 9 Prabhu, K. A., 681, (2)1 Petersen, K, (2)67, (2)69 Prabhu, K. M. M., 681 Peterson, C. W., 731, (2)248, (2)304 Prešnajder, P., (2)406 Peticolis, W. L., 8 Press, W. H., 285, 680, 699, 711, 712, 713, 730, Petranek, S., 681 (2)298 Petrashen, P. V., 449 Prestini, E., (2)41 Peyrin, F., (2)1 Prevedelli, M., (2)406 Pfaffelhuber, E., 414, (2)116 Price, P. J., (2)403 Philipp, H. R., (2)303 Price, W. C., (2)303 Phillips, K., 415 Priestley, H. A., 79 Phillips, P. S., (2)406 Priestley, M. B., (2)180 Phillips, S. C., (2)180 Pringsheim, A., 141 Phong, D. H., 413 Prise, M. E., (2)406

Author index 835

Pritz, T., (2)451 Ritter, G., 328 Privalov, I. I., 141, 316, 326, 544 Rivière, N. M., (2)41, (2)62, (2)69 Priwaloff, I., see Privalov, I. I. Robin, L., 585 Prokopenko, G. V., (2)406, (2)406 Robinson, T. S., (2)303 Prössdorf, S., 8, 632 Rodrigues, B. O., 449, (2)156 Protopopescu, V., 79 Rodriguez, E., (2)406 Protzmann, M., (2)179 Roe, D. C., (2)450 Prudnikov, A. P., 519 Roessler, D. M., (2)303 Purcell, E. M., (2)450 Rogers, L. J., 79 Pykhteev, G. N., 732 Rokhlin, V., 632, 730 Roman, P., (2)1, (2)425, (2)449 Qian, T., (2)179 Rooney, P. G., 189, 190, 193, 246, 285, 585 Qu, W., (2)406, (2)406, (2)406 Roos, B. W., 78, 79, 141, 519, 585, (2)116, (2)117 Queen, N. M., (2)250 Rosei, R., (2)406 Rosenblum, M., 585 Rabheru, A., (2)406 Rosenblum, M. G., (2)449 Rabinowitz, P. R., 730, 731 Rosenfeld, L., (2)247, (2)339, (2)340, (2)342, Radar, C. M., 680, 681 (2)347 Radon, J., 1, 277, 278, 279, 280, 281, 282, 284, Ross, G., (2)406, (2)449 285, 286, 287 Ross, W. T., 141, 328 Rakotondratsimba, Y., 681 Rossetti, G., (2)406 Ramachandran, G. N., (2)418, (2)449 Roth, J., (2)304 Raman, C. V., 8, (2)89 Rother, M., 681 Ramírez, J. G., (2)179 Roux, J., (2)450 Rao, A., (2)179 Rovnyak, J., 585 Rao, B., (2)304 Roy, R., 570 Rapapa, N. P., (2)376, (2)403 Rubio de Francia, J. L., 415, (2)61 Rasigni, G., (2)303 Rudakov, V., (2)304 Rasigni, M., (2)303 Russell, J. S., 625 Räty, J., (2)303 Ryzhik, I. M., 522 Rayleigh (Lord), 21, 462 Raynor, S., 632 Saal, L., (2)69 Read, R. R., 680, (2)178 Saarinen, J. J., (2)406, (2)406, (2)386, (2)395, Reddy, G. R., 681, (2)1, (2)178 (2)397, (2)402, (2)403, (2)404 Redheffer, R., 141, (2)44, (2)45, (2)47 Saavedra, I., (2)116 Redivo Zaglia, M., 79 Sabri, M. S., 680, (2)178 Reich, E., (2)41 Šachl, V., (2)304 Reiche, F., (2)225, (2)284, (2)324 Sack, R. A., 699, 731 Reick, C. H., (2)248 Sad, N., 519 Reid, C., 8 Sadosky, C., 414, 415 Reinhold, K., (2)41 Sagher, Y., 202, 246, 248, 346 Remacle, F., (2)89, (2)116 Saigo, M., 285 Rhodes, D. R., (2)248 Saito, M., 681 Ricci, F., (2)406 Sakai, H., 8 Richards, F. B., 681 Saksman, E., 416 Richards, P. G., 8 Samotij, K., 415 Richardson, L. F., 71, 82 Sancaktar, S., 79 Ridener, Jr., F. L., (2)403 Sandell, M., (2)179 Riemann, G. F. B., 3, 13, 25, 26, 37, 45, 46, 50, Santhanam, B., (2)180 51, 52, 61, 63, 66, 70, 128, 177, 236, 259, Santi, E., 732 287, 327, 331, 332, 543, 544, 545, 546, 549, Santini, P. M., 633 550, 606, 662, 677, 711, (2)39, (2)193, Sasaki, T., (2)406 (2)199, (2)299 Saslow, W. M., (2)240, (2)242, (2)249, (2)250 Riemenschneider, S. D., 681, (2)406 Sathyanarayana, P., 681, (2)406, (2)406 Riesz, M., 58, 141, 175, 203, 204, 205, 211, 215, Sato, R., (2)69 224, 247, 251, 254, 318, 321, 331, 333, 335, Saut, J.-C., 633 338, 340, 354, 355, 359, 364, 365, 367, 374, Savage, C. M., (2)406 379, 382, 395, 413, 497, 531, 587, 637, 661, Saxton, W. O., (2)449 (2)21, (2)22, (2)25, (2)29, (2)30, (2)38, Scaife, B. K. P., (2)240, (2)247, (2)249 (2)43, (2)44, (2)45, (2)52, (2)62, (2)234 Scandolo, S., (2)1, (2)249, (2)374, (2)376, (2)396, Rihacek, A. W., 247 (2)403, (2)404 Rimmer, M. P., (2)303 Scarfone, L. M., (2)303

836 Author index

Schafer, R. W., 8, 680, 681, (2)406,(2)178 Sidky, E. Y., 732 Schatz, P. N., 303 Siliqi, D., (2)449 Schiff, L. I., (2)116, (2)424, (2)449 Silva, H., (2)247 Schilling, M., (2)406 Silva Neto, M. J., (2)406 Schilow, G. E., see Shilov, G. E. Silverstein, M. L., 386 Schlömilch, O., 5, 8, 96 Simon, M., (2)180 Schlurmann, T., (2)180 Simpson, R. S., (2)179 Schmidt, E., 696 Simpson, T., 684, 688, 691 Schmidt, R., 248 Sindoni, O. I., (2)248, (2)303 Schmueser, P., (2)406 Singh, O. P., 248, 519, (2)36, (2)37, (2)42 Schnatterly, S. E., (2)325, (2)348 Sivashinsky, G. I., 633 Schneider, C. B., 730 Skwarek, V., (2)179 Schochet, S., 633 Slichter, C. P., (2)303, (2)348 Schoeck, G., 633 Sloan, I. H., 585, 731 Schrödinger, E. R. J. A., (2)105, (2)108, (2)363, Smallcombe, S. H., (2)450 (2)364, (2)423, (2)428 Smet, F., (2)1, (2)403 Schulz, H., (2)406 Smet, P., (2)1, (2)403 Schulze, T., 633 Smethells, G. J., 79, 717 Schützer, W., (2)105, (2)108, (2)116 Smirnoff, V., 141 Schwartz, L., 68, 69, 70, 79, (2)476, 479, 487, Smith, D. A., 79, (2)406 497, 517, 519, (2)58, (2)111, (2)113, (2)114, Smith, D. Y., (2)224, (2)241, (2)242, (2)243, (2)117 (2)248, (2)249, (2)250, (2)406, (2)260, Schwarz, H. A., 58, 205, 207, 218, 220, 287, 343, (2)267, (2)303, (2)406, (2)348, (2)349 362, 387, (2)101, (2)301 Smith, E., 632 Schwarzl, F. R., (2)451 Smith, N. I., 8 Scouler, W. J., (2)303 Smith, S. D., (2)406 Scuderi, L., 731 Smith, W. E., 8, 731 Seaton, C. T., (2)406 Smithies, F., 13, 78 Šebesta, J., (2)406 Sneddon, I. N., 8 Secrest, D., 694, 698, 704, 731 Sobolev, S. L., 70, 482, 520 Segal, I. E., (2)116 Söhngen, H., 632 Segovia, C., 285 Sokhotsky, Yu.-K. V., 111, 112, (2)41, (2)413 Seip, K., 681 Sokół-Sokółski, K., (2)21, (2)42 Selesnick, I. W., (2)179 Solov’ev, L., (2)304 Sellmeier, W., (2)248 Sommer, G., (2)179 Sen, P., (2)403 Sommerfeld, A., (2)250 Sen, P. K., (2)403 Sonine, N. J., 460 Serdyukov, A. N., (2)349 Spanos, P. D., (2)179 Sestieri, A., (2)179 Splettstösser, W., 414 Shadrin, P. M., (2)406 Srivastav, R. P., 286 Shaffer, J. C., 732 Srivastava, H. M., 632 Shaik, J. S., 8 Stallard, B. R., 8 Shamir, E., 632 Stankovi´c,B., 519 Shannon, C. E., 344, 414, (2)419, (2)420 Stanomir, D., 681 Sharnoff, M., (2)248 Stanton, C. S., 328, 411, 412, 414 Sharpley, R. C., 248, 415, (2)180 Stark, H., 247, (2)41 Shaw, R. K., 247 Steele, J. M., 80 Shea, D. F., 328, 411, 412, 414 Steenaart, W., 680, (2)178 Sheik-Bahae, M., (2)406 Stegun, I. A., 178, 268, 269, 425, 431, 453, 454, Shen, S. S. P., (2)406, (2)406, (2)180 457, 461, 463, 471, 522 Shen, Y. R., (2)403 Steiglitz, K., (2)180 Shen, Z., (2)406, (2)406, (2)406, (2)406, (2)406 Stein, E. M., 158, 241, 244, 246, 248, 413, 414, Shih, H. H., (2)406, (2)406, (2)406,(2)180 415, (2)32, (2)39, (2)41, (2)42, (2)62, (2)69 Shiles, E., (2)249, (2)303 Stein, P., 331 Shilov, G. E., 79, 516, 517, 519 Stenger, F., 414 Shirotov, V. V., (2)406 Stens, R. L., 414, 681 Shohat, J. A., (2)249 Stepantsov, E. A., (2)406 Shore, K. A., (2)403 Stephens, P. J., (2)348 Shtrauss, V., (2)247 Stern, E. A., (2)349 Shul’man, A. Ya., (2)406, (2)409, (2)449 Stern, F., (2)228, (2)240, (2)242, (2)248, (2)303 Shyu, J.-J., (2)406 Stewart, C. E., 732

Author index 837

Stieltjes, T. J., 1, 231, 232, 236, 239, 241, 265, Titchmarsh, E. C., 3, 20, 78, 79, 98, 99, 128, 141, 267, 287, 335, 364, 420, 421, 609, (2)209 170, 204, 211, 215, 219, 247, 256, 328, 414, Stirling, J., 74 518, (2)77, (2)86, (2)90, (2)91, (2)92, (2)102, Stoer, J., 706 (2)104, (2)109, (2)110, (2)112, (2)113, Stoneham, R. J., (2)304 (2)205, (2)336, (2)343, (2)401 Stout, G. H., (2)421 Tokura, Y., (2)406, (2)406 Stroud, A. H., 694, 698, 704, 731 Toland, J. F., 352, 519, 633 Stroud, D., (2)303 Toll, J. S., 247, (2)116, (2)248, (2)303 Struik, L. C. E., (2)451 Tomiyama, K., (2)248 Strutt, J. W., see Rayleigh (Lord) Tomlinson, G. R., (2)180 Struve, H., 454, 460, 461, (2)455, (2)456 Tonelli, L., 55, 56, 60, 119 Su, S. Y., (2)406 Torchinsky, A., 142, 415, (2)41 Sucher, J., 8, (2)450 Torii, T., 731, 732 Suëtaka, W., 303 Torrea, J. L., 285, 416, (2)66 Sugawara, M., (2)117 Trakhtman, A. M., 681 Sugiyama, A., 8 Trebels, W., 8 Sukhatme, U. P., (2)450 Treil, S. R., 414, 415, 416 Suzuki, N., 585 Treitel, S., 680, (2)178 Svirko, Y., (2)406, (2)404 Tricomi, F. G., 8, 79, 187, 188, 189, 190, 192, 193, Swamy, M. N. S., 681, (2)406, (2)406 195, 256, 259, 267, 276, 285, 313, 314, 330, Symanzik, K., (2)249 453, 456, 468, 469, 471, 521, 522, 529, 531, Szabo, T. L., (2)446, (2)450 543, 560, 563, 568, 569, 584, 585, 589, 602, Szegö, G., 386, 388, 414 609, 615, 632, (2)41, (2)234, (2)251 Sztaba, U., 519 Troup, G. J., 731, (2)248, (2)403, (2)450 Tschoegl, N. W., (2)451 Tabakin, F., 79 Tseng, C.-C., (2)163, (2)180 Tacer, B., (2)179 Tsereteli, O. D., 141, 414 Taft, E. A., (2)303 Tsuboi, T., (2)1, (2)406 Tagami, S., (2)179 Tuan, V. K., 247, 632 Tamarkin, J. D., 141, 211, 215, 247, (2)249 Tung, C. C., (2)406, (2)406, (2)406,(2)180 Tan, R., 202, 246, 346 Turi, J., 8 Tang, M.-T., (2)449 Turpin, P.-Y., 8 Tarasov, M. A., (2)406, (2)409, (2)449 Tuttle, Jr., D. F., (2)450 Tauber, A., 3, 131, 141, 328 Ueda, A., (2)267, (2)303 Taurian, O. E., 732 Urban, K., (2)406 Taylor, B., 32, 690, (2)96, (2)100, (2)164 Urban, M. W., (2)303 Taylor, J. G., 247, (2)112, (2)114, (2)117 Urbina, W., (2)70 Taylor, L. S., (2)449 Urciuolo, M., (2)69 Tchebichef, P. L., see Chebyshev, P. L. Urkowitz, H., 247, (2)179 Tejedor-Tejedor, M. I., (2)303 Ursell, F., 285, 431, 434, 436 Temkin, A., (2)450 Usero, D., 633 Temple, G., 65 Teng, T.-L., (2)180 Vaˇınshteˇın,L. A., (2)179 Tepichín, E., (2)179 Vakman, D., (2)132, (2)134, (2)179 Ter Haar, D., (2)247 Van Der Pol, B., 79 Terhune, R. W., (2)358 Van Der Walt, S. J., (2)349 Teukolsky, S. A., 285, 680, 699, 711, 712, 713, Van Groenendael, A., (2)403 730, (2)406 Van Kampen, N. G., (2)103, (2)105, (2)109, Thacher, Jr., H. C., 285 (2)110, (2)116, (2)117 Themistoclakis, W., 585 Van Stryland, E. W., (2)406 Thiele, C., (2)59, (2)69 Van Vleck, J. H., (2)247 Thirring, W. E., (2)117, (2)248 Van Wijngaarden, L., (2)248 Thomas, W., (2)225, (2)284, (2)324 Vanasse, G. A., 8 Thomson, W., see Kelvin (Lord) Vance, J., (2)406, (2)406 Thomaz, M. T., 335, 342, 349 Vandermonde, A.-T., 72, 73, 74 Thomée, V., 633 VanderNoot, T. J., 731 Thoone, G. P. J. M., 8, (2)451 Varela, F. J., (2)406 Tickanen, L. D., (2)303 Vargas, A., (2)61 Tinkham, M., (2)303 Varsavsky, O. A., 681 Tiomno, J., (2)105, (2)108, (2)116 Vartiainen, E. M., (2)1, (2)406 (2)406, (2)303, Tip, A., (2)449 (2)396, (2)403, (2)406

838 Author index

Vatchev, V., (2)180 Weierstrass, K. T. W., 36, 47, 48 Vaughan, R. C., 681 Weinberg, D. A., (2)406, (2)69 Vázquez, L., 633 Weinreich, G., (2)406, (2)406, (2)406 Veal, B. W., (2)303 Weir, A. J., 57, 79 Vega, L., 633 Weiss, G., 94, 141, 241, 244, 246, 247, 248, (2)32, Velický, B., (2)303 (2)41, (2)42 Veltcheva, A. D., (2)180 Weiss, M., 413 Venturi, A., 585 Weisstein, E., 471 Verbitskiˇı,I. È., 333 Weniger, E. J., 79 Verbitsky, I. È., 247, 328 Weyl, H., 661 Verdet, E, (2)310 Wheeden, R. L., 285, 321, 354, 388, 395, 397, Verdun, F. R., (2)449 399, 403, 405, 407, 414, 415, 679, 680, 683 Verleur, H. W., (2)303 Wheeler, J. C., 699 Vernon, Jr., F. L., (2)409 Wheeler, M. W. L., (2)406 Veselago, V. G., (2)250 Wheelon, A. D., 247 Vetterling, W. T., 285, 680, 699, 711, 712, 713, White, G. D., 79 730, (2)406 Whittaker, E. T., 78, 344, 414, (2)419, (2)420 Viète, F., 235, 248 Wickramasinghe, N. C., (2)248 Villani, A., (2)236, (2)240, (2)241, (2)242, (2)243, Widom, H., 171, 585, 632 (2)250, (2)251, (2)303, (2)348 Wiener, N., 6, 106, 141, 215, (2)95, (2)98, (2)116, Villarroya, P., (2)69 (2)180 Ville, J., (2)177 Wightman, A. S., (2)42 Vinti, G., 681 Wigner, E. P., (2)109, (2)117 Violini, G., (2)250 Williams, C. P., (2)410 Vitali, G., 241, 248, 382 Williams, J. R., 680 Vladimirov, V. S., (2)42 Williams, M. Z., 303 Voelcker, H. B., (2)179 Williams, W. E., 632 Volberg, A. L., 415, 416 Willsky, A. S., (2)116 Volk, M., (2)406 Wilmshurst, J. K., (2)303 Volkov, O. Y., (2)449 Wilson, G. V. H., (2)450 Volterra, V., 589 Wimp, J., 79, 732 von Neumann, J., (2)364 Witomski, P., 495, 519 Vrinceanu, D., (2)450 Witte, H., M., 681 Vystavkin, A. N., (2)406, (2)406 Wittwer, J., 416 Wohlers, M. R., 479, 519, (2)113, (2)117 Wagenaar, R. W., (2)406 Wokaun, A., (2)435, (2)406 Wagner, P., (2)406 Wolf, A. A., 285 Wainger, S., (2)62, (2)63, (2)69 Wolf, E., (2)179, (2)248, (2)348 Walden, A. T., (2)180 Wolfersdorf, L. von, 416, 632 Waldvogel, J., 731 Wong, D. Y., (2)117 Walker, J. S., 78, 680 Wong, R., 420, 422, 434, 731 Wall, H. S., (2)209, (2)249 Wood, F. M., 98, 141 Wallis, D. H., (2)248 Wood, J. K., 633 Walther, A., (2)449 Woodley, J., (2)406 Woolcock, W. S., 434 Wang, G., (2)42 Woolfson, M. S., (2)406 Wang, L. J., (2)116 Wooten, F., (2)248 Wang, P.-H., (2)180 Wright, J., (2)406, (2)406 Wang, S.-L., 328 Wright, K. R., 8 Wang, T.-C. L., (2)450 Wu, C.-H., (2)249 Wang, T.-X., (2)178 Wu, C.-K., (2)303 Ward, J. F., (2)406 Wu, M. C., (2)406, (2)406, (2)406, (2)406,(2)180 Warnock, R. L., (2)219 Wu, M.-L., (2)406 Waters, K. R., (2)446, (2)450, (2)451 Wu, T. T., (2)249 Watson, D. K., (2)406, (2)406 Wu, T.-Y., (2)116 Watson, G. N., 78, 434, 459, 465, 468 Watson, K. M., (2)116, (2)239, (2)449 Xiang, N., 248 Weaver, R. L., 8, (2)249, (2)450 Xu, Y., 247, (2)406 Weber, H. M., 453, 470, 471, (2)455 Weber, W. E., (2)179 Yamamoto, K., (2)303 Wegert, E., 416 Yamamoto, T., 731 Weideman, J. A. C., 247, 285, 723, 724, 726 Yan, D., 247

Author index 839

Yang, B.-C., 681 Zayed, A. I., 8, 414, (2)69, (2)159, (2)169, (2)180, Yang, J. N., (2)180 (2)181 Yano, S., 414 Zemach, C., (2)449 Yariv, A., (2)391, (2)392, (2)403 Zemanian, A. H., 481, 487, 495, 512, 513, 515, Yeh, H. C., (2)406 519, (2)113 Yeh, M.-H., (2)163, (2)168, (2)180 Zhang, J. E., (2)406 Yen, N.-C., (2)406, (2)406, (2)406,(2)180 Zhang, S. L., 712, 713, 732 Yeo, R. C. K., 8, (2)449 Zhechev, B., (2)178 Youla, D. C., (2)116 Zheng, D., 416 Young, R. H., (2)303 Zheng, Q., (2)406, (2)406, (2)406,(2)180 Young, W. H., 3, 84, 85, 98, 127, 132, 141, 378, Zhizhiashvili, L. V., 328, 414 379, 382, (2)53 Zhou, K., 246 Zhu, Y. M., (2)1 Ziesler, S. N., (2)69 Zabusky, N. J., 625 Zimering, S., 434 Zadeh, L. A., (2)116 Zimerman, A. H., (2)236, (2)240, (2)241, (2)242, Zahradník, J., 8 (2)243, (2)250, (2)251, (2)303, (2)348 Zaidi, N. H., (2)69 Zordan, C., 8 Žak, I. E., 328 Zweifel, P. F., 79, 632 Zalevsky, Z., (2)147, (2)148, (2)406, (2)163, Zygmund, A., 78, 141, 248, 299, 321, 328, 340, (2)406,(2)180 341, 343, 352, 363, 411, 413, 415, 681, (2)1, Zanotti, G., 681, (2)421 (2)2, (2)3, (2)9, (2)20, (2)21, (2)22, (2)41, Zaumseil, P., 449 (2)42

Subject index

Page numbers for volume 2 are indicated by (2).

Ap class, 388 product, 182–3 Ap condition, 388–94, 397 Anger functions, 470, (2)455 discrete analog, 679–80 angle of attack, 578 two-weights version, 404 anharmonic oscillator, (2)359 Abelian group, 40 anisotropic media, (2)278 Abraham–Lorentz equation, (2)93 anomalous dispersion, (2)186–7 absolutely convergent, 46 anticommutator, 156 absorption coefficient, (2)202, (2)248 Argand diagram, 24 acausal problem, (2)93 argument, 24 accumulation point, 39 argument principle, 546, 548 acoustics, (2)444 associated Legendre function of the first kind, active system, (2)73 (2)456 additivity property, (2)74 associated Legendre function of the second kind, adiabatic compressibility, (2)444 (2)456 adiabatic condition, (2)281 associative law of multiplication, 40 adjoint operator, 42 asymptotic behavior, admittance, (2)438, (2)442 conductance, (2)267 analytic structure, (2)438 dielectric constant, (2)185–6 crossing symmetry relation, (2)439 dielectric tensor, (2)280 dispersion relations, (2)439 dispersion relation, (2)219–22 airfoil equation, 522 elastic scattering amplitude, (2)426 airfoil problem, 577–82 energy loss function, (2)270 generalized, 582 harmonic generation susceptibility, (2)361, Akhiezer’s theorem, 200 (2)372–6 algebra, (2)64 magnetic susceptibility, (2)273–4 σ -algebra, (2)64 natural optical activity, (2)340–2 allied series, 302 nonlinear susceptibility, (2)372–6 almost all, 53 optical rotatory dispersion, (2)340–2 almost everywhere, 53 phase, (2)260–1 amplitude modulation (AM), (2)135–8 reflectivity, (2)255, (2)258, (2)260 analog signals, (2)119 refractive index, (2)204, (2)207 analytic, 25 N+ and N−, (2)316 analytic function, 25, (2)38 Rosenfeld formula, (2)340 discrete, 675 surface impedance, (2)278 analytic operator, (2)127 asymptotic expansions, 419 analytic part, 32 of one-sided Hilbert transform, analytic signal, 181, (2)126–35, (2)234 422–34 eigenfunction of the Hilbert transform operator, of Stieltjes transform, 420–2 (2)131 atlas of Hilbert transforms, (2)534–46 Fourier transform, (2)128, (2)138 atomic units, a.u., (2)423, (2)432 fractional, (2)169 attenuation coefficient, (2)444

840

Subject index 841

dispersion relation, (2)446 bounded doubling “time,” (2)63 auto-convolution function, (2)123 bounded operator, 157 auto-correlation function, (2)123 bounded set, 39 normalized, (2)126 bounded variation, 232 automorphism, 45 bra, 41 Alzer’s inequality, 665 branch cut, 27 branch line, 27 band-limited function, 345, 674, (2)135 branch point, 26 band-pass filter, (2)120 band-stop filter, (2)120 Cn, symmetry operation, (2)310 Basset’s function, (2)456 (C,1) summable series, 301 Calderón–Zygmund inequality, (2)21–4 Bedrosian’s theorem, 184–7, 455, (2)131 Calderón–Zygmund operators, (2)2 discrete Hilbert transform, 674 truncated, (2)2–3 Benjamin–Ono equation, 624–30 Calderón–Zygmund singular operator, (2)2 conservation laws, 627–30 camber line, 577 linearized form, 633 cancellation behavior for the Hilbert transform, Bernstein’s inequality, 343 135–40 Bessel functions (modified) 465, (2)455–6 Carleman’s approach, finite Hilbert transform, Bessel functions (spherical), 462 550–2, 606–8 Hilbert transform, 462–4 carrier frequency, (2)136 Bessel functions of the first kind, 187, 453–9, cas function, 262 (2)455 Catalan’s constant, 338, (2)457 for non-integer index, 459–60 Cauchy kernel, 107 Hilbert transforms, 460 Cauchy integral, 85, 107–13 Fourier transform of J0(ax), 453 ( ) Cauchy integral formula, 30 Hilbert transform of J0 ax , 453–54 Cauchy integral theorem, 29 Hilbert transform of sin ax Jn(bx) and ( ) Cauchy principal value, 2, 13–4, 58 cos ax Jn bx , 456–9 Cauchy principal value integrals, 13–4, 58 Bessel functions of the second kind, 453, (2)457 elementary transformations, 684–8 for non-integer index, 459–60 Cauchy pulse, 4 Hilbert transforms, 460 Cauchy residue theorem, 33–4 Bessel functions of the third kind, 453 Cauchy sequence, 42–3 Bessel’s differential equation, 453 Cauchy transform, 85 Bessel’s inequality, 19 Cauchy–Goursat theorem, 29 beta function, 271, (2)454 Cauchy–Riemann equations, 25, (2)39 bijective, 44 Cauchy–Schwarz–Buniakowski inequality, 58 bilateral Laplace transform, 267 causal condition, 649, (2)77, (2)80–1, (2)105 bilinear Hilbert transform, (2)58–9 Schützer and Tiomno, (2)108 bilinear transformation, 38 Van Kampen, (2)109 binomial coefficient, 74 causal distributions, 480 Blaschke product, (2)255, (2)412 causal function, 215, 265 Bloch equations, (2)273–4 causal sequence, 652 Boas transform, (2)49–57 Fourier transform, 656 Fourier transform, (2)53 causal system, (2)81 generalization, (2)56 causal transform, 216, (2)104–5 inversion of, (2)55 causality, (2)80–1, (2)102–5, (2)190, (2)192, iteration formula, (2)52 (2)194, (2)197, (2)207–8, (2)250, (2)336, Parseval-type formula, (2)51 (2)343, (2)353 relation to Hilbert transform, (2)49 Einstein, (2)208 Riesz-type bound, (2)52 extensions, (2)102 Boas’ theorem, (2)54 principle of relativistic, (2)207 Boas–Heinig transform, (2)57 Cayley transformation, 38 Bochner’s theorem, (2)83 central slice theorem, 281 Bode relations, 6 Cesàro summable series, 301 Boltzmann distribution, (2)362 characteristic function, 42, 51, 379 Borel field, 55 characteristic value, 42 Borel measure, 55 charge density, (2)280 Borel set, 55 Chebyshev polynomial, 439, 452 Born approximation, (2)425, (2)429 finite Hilbert transform, 567–70 exchange term, (2)429 Chebyshev polynomial of the first kind, (2)457 analytic structure, (2)432 Chebyshev polynomial of the second kind, (2)457

842 Subject index

Chebyshev’s inequality, 365 conservation laws, 627–30 chemical exchange processes, (2)436 constant of motion, 627 circular birefringence, (2)306 contour, 27–9 circular dichroism, (2)307, semicircular, 31, 86 (2)335, (2)340–2, (2)344–6 convergence, circular Hilbert transform, 293 accelerator, 70, 717–8, 723 circular polarization, (2)307–9 pointwise, 54 left-circularly polarized, (2)309 uniform, 46 right-circularly polarized, (2)309 convergent series and integrals circulation density, 578 absolutely, 46 Clausen function, (2)455, (2)507 conditionally, 46 Clausius–Mossotti equation, (2)339 convolution property, 167–9, 307–8, 645 closed contour, 27 for distributions, 482, 491–5, (2)33, (2)35 closed interval, 13 convolution theorem, 21 closed set, 39 correlation identity, (2)124 closure theorem, (2)340 cosine integral function, Ci(x), 187, 470, 522, cofinite Hilbert transform, 582–4 (2)454 Cole–Cole plot, (2)436 Hilbert transform, 470 commutator of two operators, 42, 156 cosine-exponential integral, (2)454 of singular integral operators, (2)59 cotangent form, 527–9 compact operator, 45 Cotlar’s inequality, 374 compact set, 45 Cotton effect, (2)335 compact support, 68 Cotton–Mouton effect, (2)306 complement of sets, 39 countable set, 49 complementary error function, 276 covering lemma, 241, 382 complete, 18 cross-correlation function, (2)124 complex conjugation property, 145 cross-section, (2)89, (2)105 complex dielectric constant, (2)182, (2)231 differential elastic scattering, (2)247 complex optical rotation function, (2)321, (2)335, total scattering, (2)245, (2)429 (2)339, (2)343–4 crossing condition, S-function, (2)103 complex reflectivity, (2)252 complex refractive index, (2)201–8, (2)236 crossing symmetry relation, (2)85, (2)187, (2)189, (2)193, (2)201, (2)206, (2)210, N+ and N−, (2)309–16 (2)244, (2)306, (2)313, (2)356, (2)381, complex wave number, (2)444 (2)383 condensation, (2)444 admittance, (2)439 conditionally convergent, 46 conductance, (2)267 electric susceptibility, (2)187, (2)189 asymptotic behavior, (2)267 dielectric constant, (2)193 circuit, (2)440 dielectric tensor, (2)279, (2)281 dispersion relations, (2)268–9 forward scattering amplitude, (2)244 conducting materials, (2)216–9 frequency domain compressibility, (2)444 Kramers–Kronig relations, (2)218 generalized electric susceptibility, (2)356, singularity at the zero frequency, (2)218 (2)383 conformal mapping, 37–9, (2)208, (2)290–1 harmonic generation susceptibility, (2)383 conjugate Fourier series, 129, 288, 296, 299, inverse dielectric function, (2)283 301–2, 323, 684, 713–7, (2)290 magnetoreflectivity, (2)327 dielectric constant, (2)291–3 modulus of elasticity, (2)448 reflectance, (2)294–8 nonlinear electric susceptibility, (2)356 conjugate function, 115 nonlinear refractive index, (2)400 conjugate Poisson formulas, 88 optical rotation, (2)344 conjugate Poisson kernel, 88, 339, (2)48 permeability, (2)272 conjugate Poisson operator, upper half plane, reflectivity, (2)255, (2)257 374–5 circular polarized, (2)327 conjugate series, 129, 288, 296, 302, 309–10, 326 refractive index, (2)206 conjugation (complex) property, 145 N+ and N−, (2)313, (2)344 conjugation operator, 130 surface impedance, (2)277 continuous functions, current density, (2)280 Hölder, 12, 45 current operator, (2)337 piecewise, 15 current–current response tensor, (2)280, (2)337 sectionally, 110 Cusmariu’s definition of the fractional Hilbert continuous operator, 204 transform, (2)160–3

Subject index 843

Cusmariu’s definitions of the fractional analytic inverse, IDFT, 640, 642 signal, (2)169–70 one-sided, 640 cyclotron frequency, (2)314 Parseval’s relation, 641 properties, 640–1 damped harmonic oscillator, (2)75–8, (2)87 two-sided, 640 Dawson’s integral, 257 discrete fractional Fourier transform (DFRFT), delta distribution, 63–5, 68 (2)163–8 delta function, 63 discrete fractional Hilbert transform (DFHT), demodulation, (2)136 (2)168–9 density, 107, 228 discrete Hermite–Gaussian functions, (2)163–5 density fluctuation operator, (2)281 discrete Hilbert transform, 637, 656, 660–75, density function, 107 (2)169 density matrix, (2)362 Bedrosian’s theorem, 674 density operator, (2)362 inversion formula, 669, 672–3 matrix elements, (2)363 isometric property, 671–2 density–density response function, (2)281 Parseval-type identity, 671, 674 derivative of Hilbert transform, 164 weighted inequalities, 679–80 derivative property (distributions), 518 DISPA analysis, (2)435–7 DFT, see discrete Fourier transform dispersion, (2)182 dielectric constant, (2)183 anomalous, (2)186–7 asymptotic behavior, (2)185 normal, (2)186 conjugate Fourier series, (2)290–4 dispersion formula, (2)182, (2)244, (2)248 Kramers–Kronig relations, (2)184–201 Lorentz, (2)239, (2)244 sum rules, (2)188, (2)222–7, (2)231–6, dispersion relations, 7, (2)102, (2)105 (2)239, (2)284–5, (2)287–90, (2)299 admittance, (2)439 table, (2)240–1 asymptotic behavior, (2)219–22 dielectric tensor, (2)278 attenuation coefficient, (2)446 analytic behavior, (2)279 conductance, (2)268–69 asymptotic behavior, (2)280, (2)316 dielectric constant, (2)189 crossing symmetry, (2)279, (2)281 dielectric tensor, (2)279–80, (2)287–9 dispersion relations, (2)279–80, (2)282–3, elastic scattering amplitude, (2)428 (2)287–9 electric susceptibility, (2)189 sum rules, (2)287–90 electron–H atom scattering, (2)429 symmetry properties, (2)279, (2)289 ellipticity function, (2)323, (2)345–6 difference frequency generation, (2)358 energy loss function, (2)283 difference of sets, 39 forward scattering amplitude, (2)245–6 difference operator, 74 harmonic generation susceptibility, (2)377–84 differential elastic scattering cross-section, impedance, (2)441–2 (2)247 magnetic optical activity, (2)319–23 digamma (psi) function, 178, (2)457 magnetic susceptibility, (2)273, (2)433 digital signals, (2)119 magnetoreflection, (2)325–30 dilation operator, 156, (2)15 magneto-rotatory, (2)320–1, (2)323 dilogarithm function, (2)456, (2)507 modulus of elasticity, (2)448 dipole moment, (2)182, (2)363, (2)367 nonlinear electric susceptibility, (2)377, Dirac bra–ket notation, 41 (2)386–7 Dirac comb distribution, 512, 682 cases where they do not apply, (2)388 Dirac delta distribution, 63–5, 68–9, 474–7, 483, experimental verification, (2)384 485, 491, 514 n-dimensional form, (2)387 Fourier transform, 488 two-variable form, (2)386 Hilbert transform, 491 nonlinear refractive index, (2)400, (2)402 support, 485 normal incident reflectance, (2)257–63 direct product of distributions, 492 optical activity, (2)330–48 directional Hilbert transform, (2)60–1 permeability, (2)271–4 directional maximal Hilbert transform, (2)61 phase, (2)257–63, (2)385 Dirichlet conditions, 17 potential scattering, (2)425–8 Dirichlet kernel, 296, 298 refractive index, (2)206 discrete Ap condition, 679–80 single subtracted, (2)211 discrete analytic functions, 675–6 subtracted, (2)211, (2)384 discrete Fourier transform, DFT, 637–44 surface impedance, (2)277–8 centered, 640 distributions, 63 evaluation, 641–3 analytic representation, 498–502

844 Subject index

distributions (cont.) of the Fourier transform, (2)155 bounded from the left, 495 of the Hilbert transform operator, 195–9, 311–2 bounded from the right, 495 of the n-dimensional Hilbert transform, (2)17–8 causal, 480 of the one-sided Hilbert transform, 612, 617 convolution operation, 482, 491–5 Einstein causality, (2)208 + − δ , δ , 475–6, 486 Einstein convention (in tensor analysis), (2)279 inverse Fourier transform, (2)115 elastic scattering amplitude, (2)425 derivative, 478–80, 482, 504–5, 517 dispersion relation, (2)428 Dirac comb, 512, 682 forward, (2)425 Dirac delta, 63–5, 68–9, 474–8, 480–1, 483–6, electrical circuit analysis, (2)437 488–93, 502, 507, 512–4 electric displacement, (2)183, (2)190, (2)192, (2)206, (2)278, (2)312 direct product, 492 spatial-dependent, (2)281 Fourier transform, 482, 487–90 electric field, (2)183 generalized Cauchy integral representation, electric permittivity tensor, (2)279 499 electric polarization, (2)182–3, (2)185–6, (2)315, Heaviside distribution, 485 (2)352–6, (2)360–1, (2)367, (2)398 Heisenberg delta distributions, 475–6, 486 units, (2)183 Hilbert transform, 490, 495–8, 501 electric susceptibility, (2)183, (2)187, n-dimensional, (2)5, (2)35, (2)190, (2)193–5, (2)351–2, (2)354–6, (2)37 (2)360–1, (2)367–99 periodic distributions, 508–15 analytic behavior, (2)187 Hilbert transform, 515–6 crossing symmetry, (2)187, (2)189, (2)193 −1 p.v. (x ), 69, 476–8, 483–6, 488–93, 495–8, Hilbert transform relations, (2)188 502–4, 507–8 − Kramers–Kronig relations, (2)189 P(x 1), 475–6 electron density, (2)417–8 regular, 69 electron–atom scattering, (2)422 Schwartz, 68 dispersion relations, (2)429 Schwartz’s theorem, (2)111, (2)113 electron–H atom scattering, (2)428–32 singular, 69 exchange term, (2)429 singular integral equations involving, 630–1 analytic structure, (2)429–32 slow growth, 481 ellipticity function, (2)320, (2)334 Taylor’s theorem, (2)112 dispersion relations, (2)321, (2)323, (2)345–6 temperate, 481 per unit length, (2)334 tempered, 481 empirical mode decomposition (EMD), (2)170–8 tensor product, 491–5, (2)32, (2)35 algorithm, (2)171–2 Titchmarsh’s theorem extension, (2)110–5 sifting, (2)173 distribution functions, 232–3, 338, 364 empty set, 39 distributive law, 40 energy, (2)87, (2)103, (2)106, (2)120, (2)125, (2)162, (2)423, (2)425–9 dot product (distributions), 510 energy loss function, (2)269–71 double factorial function, 162, 440 asymptotic behavior, (2)270 double Hilbert transform, (2)8–10, (2)21 sum rules, (2)270–1 double sideband modulation, (2)136 entire function, 34, (2)417 doubling property, (2)63 order, 36 Drude model, (2)223 type, 36 duality argument, 209, 334, 368 envelope function, (2)132 duplication formula (gamma function), 283 ergodic Hilbert transform, (2)63–6 dynamic form factor, (2)285 ergodic hypothesis, (2)63 dynamic stress, (2)447 ergodic maximal Hilbert transform, (2)65 ergodic theory, (2)63 error estimates (Gaussian quadrature), 706–7 eigenfunctions, 42 error function, 257, 719, (2)455 of the finite Hilbert transform, 563 essential singularity, 25 of the Fourier transform, (2)155 essential supremum, 54 of the Hilbert transform operator, 195–9, 311–2 essentially bounded functions, 54 of the n-dimensional Hilbert transform, (2)17–8 Euclidean space, 49 of the one-sided Hilbert transform, 612, 615 Euler’s constant, 522, (2)457 eigenvalue problem, 42 Euler’s integral of the first kind, (2)454 eigenvalues even Hilbert transform operator, 147, 258 of the finite Hilbert transform, 563 on the disc, 303–4

Subject index 845

expectation value, (2)363, (2)367 Fourier integral formula, 23 exponential integral, E1(x), 268 Fourier series, 14 En(x), 728, (2)455 Fourier series approach (conjugate functions), exponential integral function, Ei(x), 269, 451, 129–31 (2)455 partial sums, 296, 299, 301, 323–5, 327 period, 15 f sum rule, (2)225, (2)228, (2)230, (2)246, periodic property, 14–5 (2)250, (2)268, (2)270, (2)299, Fourier sine integral formula, 23 (2)301 Fourier sine transform, 1, 256, 259, 276 Faraday effect, (2)306–7, (2)309, (2)319–24 Fourier spectroscopy, (2)409 fast Fourier transform, 711 Fourier spectrum, (2)131, (2)135 Hilbert transform, 712 Fourier transform, 1, 19, 252 Fatou’s theorem, 94, 320, 326 alternative definitions, 20 Fejér kernel, 296, 299 analytic signal, (2)128 figure of merit, (2)422 of Bessel function J0(ax), 453 filter, (2)120 Boas transform, (2)50–6 band-pass, (2)120 of a causal sequence, 656–60 band-stop, (2)120 connection with Hilbert transform, 158, high-pass, (2)120 252–8 Hilbert, 650, (2)121–3 convolution theorem, 21 Hilbert transformer, (2)121 of Dirac delta distribution, 488 low-pass, (2)120 discrete, 637–44 ◦ 90 phase shifter, (2)121 discrete fractional (DFRFT), (2)163–8 quadrature, (2)121 distributions, 487–90 finite Hilbert transform, 443, 452, 521–82, 601–9, eigenvalues, (2)155 685, 727–30 eigenfunctions, (2)155 airfoil problem, 577–82 fast, 711 generalized, 582 of finite Hilbert transform, 557–8 of the Chebyshev polynomials, 567–70 fractional, (2)149–59 contour integration approach, 570–7 properties, (2)158–9 cosine form, 523–7 of Gaussian, 256, (2)26–7 cotangent form, 527–9 n-dimensional, (2)27 inversion by Fourier series approach, 541–3 of Heaviside distribution, 490 inversion formula, 529–43, 550–2 of Hilbert transform, 252–8 of Legendre polynomials, 563–7 distributions, 505–8 moment formula, 530–1, 554–5 inversion of one-sided Hilbert transform, 612–4 numerical evaluation, 685, 727–30 n-dimensional, (2)12–3 properties, 552 n-dimensional Hilbert transform, (2)12–4 convolution, 556–7 of p.v.(1/x), 489 derivative, 555–6 notation, 20 eigenfunctions, 563 numerical integration, 709–11 eigenvalues, 563 Parseval formula, 21 even–odd character, 552 Plancherel formula, 21 Fourier transform, 557–8 of Riesz transform, (2)28–9 inversion, 552–3 same symbol convention, (2)190 orthogonality, 562 of sgn distribution, 489–90 Parseval-type identities, 559–61 singular integral equation, 599–601 of product xnf (x), 554–5 tempered distribution, 481 scale changes, 553–4 two-dimensional, 280–1 first Born approximation, (2)425 of unit distribution, 489 floor function, 444, (2)172–3, (2)457 of the unit step sequence, 657 forward scattering amplitude, (2)239, (2)244–7 Fourier transform spectroscopy, (2)434 crossing symmetry condition, (2)244 fractional analytical signal, (2)169–70 dispersion relations, (2)244–6 Cusmariu’s definitions, (2)169–70 Fourier allied integral, 124, 275, (2)298–9 Zayed’s definition, (2)169 Fourier conjugate series, 129, 296 fractional Fourier transform (FRFT), (2)149–59 Fourier cosine integral formula, 23 index additivity property, (2)154 Fourier cosine transform, 1, 256, 259 properties, (2)158–9 Fourier integral, 22 fractional Hilbert filter, (2)148 Fourier integral approach to the Hilbert fractional Hilbert transform, (2)147, (2)168 transform, 122–8 Cusmariu’s definition, (2)160

846 Subject index

fractional Hilbert transform (cont.) hypergeometric 2F1, 203 Lohmann–Mendlovic–Zalevsky incomplete gamma, 425 definition, (2)147 integral, 34 Zayed’s definition, (2)159 Kummer’s confluent hypergeometric, 257 Fredholm equation of the first kind, 2, 589–91 maximal Hilbert transform, 373–86 Fredholm equation of the second kind, 589, 592–4 meromorphic, 30 homogeneous, 592 multiple-valued, 25 Fredholm equation of the third kind, 589, 594–9 notation, 2 free induction decay, (2)434 orthogonal, 18 free-electron model, (2)184, (2)186, (2)190 orthonormal, 18 free-electron-like system behavior, (2)224 piecewise continuous, 15 frequency rectangular step, 442–3, (2)143 carrier, (2)136 Riemann zeta, 662 cyclotron, (2)314 S, (2)105–6, (2)108–9 instantaneous, (2)132 signum (sgn), 6, (2)142 Larmor, (2)273 simple, 52 negative, (2)128, (2)187 sinc, 185, 719, (2)145, (2)419 notation, (2)76, (2)83–4 sine integral, 431, 470, 522 plasma, (2)185 spherical Bessel, 462–4 frequency domain, (2)81–3, (2)138 Struve’s, 454, 460 frequency modulation (FM), (2)135 support, 68 Fresnel cosine integral, 719, (2)454 test, 68 Fresnel sine integral, 719, (2)456 of rapid decay, 480 front velocity, (2)207 triangular, 186, (2)145 Fubini’s theorem, 55–7 unit impulse, 63 functional, 68, 70 unitary, 509 functional analysis, 39 Weber’s, 453, 471 functions, types of, gamma function, 162, 257, 271, 283, 425, (2)25, Anger, 470 (2)27–8 band-limited, 345 duplication formula, 283 Bessel, of the first kind, 187, 453–9 incomplete, 425 Bessel, of the second kind, 453, 460 reflection formula, 469 Bessel, of the third kind, 453, 464 gap Hilbert transform, 251 beta, 271 Gauss’ hypergeometric function, (2)455 cas, 262 Gauss–Legendre quadrature, 694, 697, 728 causal, 215–6, 265 Gaussian, 166, 256, 446 characteristic, 42, 51, 379 Fourier transform, (2)26–7 complementary error, 276 Gaussian quadrature, 691–4 conjugate, 115 log quadrature, 694–701 cosine integral, 187, 431, 470, 522 application to Hilbert transform, 701–6 density, 107 application to Kramers–Kronig relations, distribution, 232 708–9 entire, 34 error factors, 706–7, 744 envelope, (2)132 weights and abscissas, 735–43 error, 257, 719 specialized, 684, 695, 701 essentially bounded, 54 Gegenbauer polynomial, 439, 452, 472, floor, 444, (2)173, (2)457 (2)455 gamma, 162, 257, 271, 283, 425, (2)25, generalized function, 63, 65, 68, 475 − (2)27–8 generalized function P(x n), 475 generalized, 63, 65, 68, 475 generalized Hilbert transform, (2)2 generalized step, 52 generalized refractive index, (2)182 generating, 439–40 generalized step function, 52 Green, (2)424–5 generating function, Hankel, 453, 464–5 Legendre polynomials, 439 Hardy–Littlewood maximal, 368–73, 379–82, good λ inequality, 381 389 Gram–Schmidt orthogonalization, 696 Heaviside step, 450, (2)139 Green function, (2)424–5 Herglotz, (2)208–16 Green’s theorem, 331, 333 Hermite–Gaussian, (2)154 group, 40 discrete, (2)163 Abelian, 40 hyperbolic sine integral, 728 group velocity, (2)228

Subject index 847

half-Hilbert transform, 609 Bessel functions (modified) 465–9 Hankel functions, 453, 464–5 Bessel functions (spherical), 462–4 Hamiltonian, (2)362 bilinear, (2)58–9 two-electron, (2)428 bounded operator, 157, 203 Hardy space, 43, 99, 115 on the circle, 84, 91, 94, 114–8, 288, 292–6, Hardy’s result (finite Hilbert transform), 527 303, 311–3 Hardy–Poincaré–Bertrand formula, 57–61, 107, circular, 292, 295 112, 119, 148, 192, 223–5, 313–5, 529, 533, cofinite, 582–4 553 commutator relations, 156, 160 Hardy–Littlewood maximal function, 368–73, complex conjugation property, 145 379, 382 conjugate Fourier series evaluation, 713–7 harmonic conjugate, 87 of a constant, 102 harmonic correspondence condition, (2)134 continuous operator, 204 harmonic function, 87 convolution property, 167–9, 307–8 harmonic generation susceptibility, (2)352, cosine integral function, Ci(x), 470, (2)454 (2)357–8 derivative, 164–7 crossing symmetry, (2)381, (2)383 distributions, 504–5 dispersion relations, (2)377, (2)385–7 Dirac delta distribution, 490–1 harmonic oscillator, damped, (2)75–8, (2)87 directional Hilbert transform, (2)60–1 Hartley cas function, 262, (2)454 of the Dirichlet kernel, 296, 298–9 Hartley transform, 262 discrete, 637, 649–52, 660–80 Hartley cas function, 262 discrete fractional (DFHT), (2)168–9 of Hilbert transform, 262–5 discrete time signal, 649–52 Heaviside step function, 450, (2)139 distribution, 474–8, 490–1, 495–8 Heaviside distribution, 483, 485 inversion formula, 502–4 Fourier transform, 490 periodic, 515–6 Heisenberg delta functions (distributions), 475–6, double, (2)8–10, (2)21 486, (2)431 eigenfunctions of the Hilbert transform inverse Fourier transform, (2)115 operator, 195–9, 311–2 helical Hilbert transform, (2)66–7 eigenvalues of the Hilbert transform Helmholtz equation, (2)201 operator, 195–9, 311–2 Helmholtz wave equations, (2)276 electric susceptibility, (2)188 Helson–Szegö theorem, 386–8 ergodic, (2)63–6 Herglotz functions, (2)208–16 even functions, 146–7 Hermite polynomial, 439, (2)455 even Hilbert transform operator, 147, 258 Hilbert transform (with Gaussian on the disc, 303–4 weight), 446–9 extensions, (2)44–9, (2)58–68 orthogonality condition, 448 fast Fourier transform evaluation, 712 recurrence relation, 447 of the Fejér kernel, 299–301 Hermite’s differential equation, 446 finite, 443, 452, 521–82 Hermite–Gaussian functions, (2)154 cosine form, 523–7 discrete, (2)163–5 inversion formula, 529–43, 550–2 Hermitian operator, 42, (2)340 Fourier integral approach, 122–8 high-pass filter, (2)120 of a Fourier transform, 261 Hilbert amplitude spectrum, (2)178 Fourier transform connection, 252–8 Hilbert filter, 650, (2)121 distributions, 505–7 Hilbert problem, 544 fractional, (2)147 Hilbert space, 42 Cusmariu’s definition, (2)160 Hilbert spectroscopy, (2)406 Lohmann–Mendlovic–Zalevsky definition, Hilbert spectrum, (2)178 (2)147 Hilbert transfer function, (2)121 Zayed’s definition, (2)159 Hilbert transform, 1–2, 83–5 functions of the class L1, 211–4 ∞ allied Fourier integral evaluation, 712–3 functions of the class L , 214–5 along curves, (2)62–3 gap, 251 alternative definition, 3 Gaussian, 166, 256–7, 448 analytic signal, 181 Gaussian quadrature, 701–6 Anger function, 470–1 generalized, (2)2 atlas, (2)534–46 half-, 609 Bessel function J0(ax), 453–4 Hankel functions, 464 of sin ax Jn(bx) and cos ax Jn(bx), 456–9 Hartley transform of, 262–5 for non-integer index, 460 helical, (2)66–7

848 Subject index

Hermite polynomials (with Gaussian weight), skew-symmetric character, 147–8 446–9 spherical Bessel functions, 462–4 history, 3, 6 of a square wave, 289–91 Huang transform, (2)170–8 step functions, (2)146 inverse operator, 149 in terms of Stieltjes transform, 265–7 inversion property, 148–9, 294, 301–3, 315, Struve function, 460–2 323, 502–4, (2)10–1, (2)214 subtracted, 685, (2)114 iteration property, 148 superhilbert, (2)70 ∞ Kober’s extension for the L case, (2)47–9 table, (2)453–533 − Laguerre polynomials (with weight H(x) e x), tempered distributions, 517 449–52 Titchmarsh (Parseval-type) formulas, 170–1 Legendre polynomials, 438–46 truncated, 83, 97, 226, 253, 521, 526 linear scale changes, 150 cotangent form, 527–9 linearity, 145–6 ultradistributions, 516–18 Lipschitz condition, 98, 316 of the unit triangular pulse, (2)147 Lorentzian profile, 704–5 unitary property, 174 maximal, 373–4, 381–2 vectorial Hilbert transform, (2)60 ergodic, (2)65 weak-type (1, 1), 338, 364–8 n-dimensional, (2)10 Weber’s functions, 471 Mellin transform of, 269–75 weighted norm inequalities, 354–63, 395–408 mixed, 661 Young’s form, 84, 98 moment formula, 160–4, (2)213 Hilbert transform spectroscopy, (2)406 n-dimensional, (2)5–8, (2)10–8 absorption enhancement, (2)410–1 derivative, (2)11–2 Josephson junction, (2)406–10 Fourier transform, (2)12–4 Hilbert transformer, (2)121 product of one-dimensional operators, (2)8 Hilbert’s formula, 305 nonlinear scale transformations, 150 Hilbert’s inequality, 661–6 notation, 2–3 Hilbert’s integral, 335 numerical evaluation, 684–8, 701–6 Hilbert’s integral formula, 96, 117 by eigenfunction expansion, 723–7 Hilbert–Huang transform, (2)170–8 odd functions, 146–7 Hilbert–Stieltjes transform, 231–41, 364–5 odd Hilbert transform operator, 147, 258 hilbertogram, (2)409 on the disc, 303–4 Hölder condition, 13, 98 one-sided, 420, 422, 609–11 Hölder continuous function, 45, 59–61, 98, 107, inversion formula, 611–2 111 operator definition, 4 Hölder index, 13 orthogonality property, 174–5, 310–1 Hölder order, 13 Hölder result (for product of functions in the of oscillatory functions, 717–23 p q) pair, 5, 83 classes L and L , 54 parity property, 146 Hölder’s inequality, 57, 81–2, 96 Parseval-type formulas, 170–1 holomorphic, 25, (2)39 homeomorphic, 44 periodic function, 91, 114–20, 132–4 homeomorphism, 44, (2)37 of the Poisson kernel for the disc, 296–8 homogeneity condition, (2)134 of a product, 181–7 homogeneous condition, 355, (2)2 product of analytic signals, 182–3 homogeneous Riemann–Hilbert problem, 544 product theorem, 184 index, 545 of pulse forms, (2)146–7 homothetic operator, 156 p.v.(1/x), 475, 491 Hunt–Muckenhoupt–Wheeden theorem, 395–9, on the real line, 94–9, 120–2 679–80, 683 reciprocal relations, 148 hyperbolic sine integral function, 728, (2)457 of the rectangular pulse, (2)146 hypergeometric function F , 203 Redheffer extension, (2)44–7 2 1 reduced, 609 idempotent, 199 refractive index, (2)205 identity element, 40, 42 relationship with the Laplace transform, 267–9 IDFT, see inverse discrete Fourier renormalized, (2)47 transform scale changes, 150–5, 304–5 image point, 86 semi-discrete, 661 imaginary unit i, 23–4 semi-infinite, 609 immittance, (2)438 series expansion approach, 177–81 impedance, (2)438, (2)440–3 of the sinc pulse, (2)147 dispersion relations, (2)441–3

Subject index 849

incomplete gamma function, 425, (2)457 inverse discrete Fourier transform (IDFT), 639–40 index of a function, 546–50 inverse Hilbert transform operator, 149 index of Hölder condition, 13 inverse Mellin transform, 270 index additivity property (FRFT), (2)154 inverse point, 89 inequality, types of inversion formula, 148–9, 301–3, 590, (2)10–1, Alzer, 665 (2)17, (2)36, (2)38, (2)41, (2)213–4, Bernstein, 343–9 (2)234 Bessel, 19 Boas transform, (2)55–6 Calderón–Zygmund, (2)21–4 cofinite Hilbert transform, 583 Cauchy–Schwarz–Buniakowski, 58 discrete Fourier transform, 637–44 Chebyshev, 365 discrete Hilbert transform, 672–3 Cotlar, 374 distributions, 478, 502–4, 515, 518, (2)36–7 good λ, 381 finite Hilbert transform, 529–43, 550–2, 601–9 Hilbert, 661–6 one-sided Hilbert transform, 609–11 Hölder, 57, 81–2, 96 n-dimensional Hilbert transform, (2)10–1, Kolmogorov, 335–40, 364, 367, 408 (2)36, (2)38 Minkowski, 205–6 inversion property on the circle, 294, 301–3, relative distributional, 381 313–6 reverse Hölder, 389–90 invertible function, 44 Riesz, 203–11, 318–23, 331–4, (2)21, (2)38, isogonal transformation, 37 (2)62, (2)234, (2)236 isolated point, 25 strong-type, 364, 403 isometric operator, 43, 157, 672 triangle, 41 isomorphic vector space, 40 weak-type, 338, 364–8 isomorphism, 45, (2)38 weighted norm, 354–63, 395–408 iteration property, 148, (2)10–1 Young, 378, (2)53 Jacobi function of the second kind, (2)456 Zygmund, 340–3, 363, (2)21 Jacobi polynomial, 439, 452, 472, (2)456 infimum, 36 Jacobian, 37 inhomogeneous Riemann–Hilbert problem, Jensen’s formula, (2)97, (2)99–100 544 Jordan’s lemma, 30 inhomogeneous singular integral equation Josephson junction, (2)406–10 for H , 614–17 1 jump discontinuity, 15 injective, 44 inner product, 40 kernel function, 2 instantaneous amplitude, (2)132, (2)177 Kerr effect, (2)306 instantaneous energy, (2)178 ket, 41 ∞ instantaneous frequency, (2)132, (2)177 Kober’s extension for the L case, (2)47–9 instantaneous phase, (2)132, (2)177 Kober’s theorem, 212 integrable function, 19, 51–3 Kolmogorov’s inequality, 335–40, 364, 367, 408 Lebesgue, 51–5 weak-type, 364 locally, 69 weighted version, 363 Riemann, 45–8, 50–1 Kolmogorov’s theorem, 338, 364 integral, types of Korteweg–de Vries equation, 624, 627 Cauchy, 85, 107–14 Kotel’nikov’s theorem, see Hilbert, 335 Whittaker–Shannon–Kotel’nikov theorem Poisson, for the half plane, 85–9 Kramers–Kronig relations, 6, 708, (2)182, Poisson, for the disc, 89–94 (2)184–201 integral equation, 2, 588 absorption coefficient, (2)207 integral function, 34 admittance, (2)439–40 integral inequalities, optical constants, assumptions, (2)190, (2)225, (2)239 (2)300–2 causality, (2)190, (2)192, (2)194, (2)197, integration (2)207–8, (2)250 Lebesgue, 52 conductance, (2)268–9 Riemann, 45–6 conducting materials, (2)218, (2)267 intensity reflectivity, (2)253 derivation, direct on the interval [0, ∞), interpolation theorem, 365, 372–3, 404 (2)199–201 interval derivation, rigorous, (2)190–7 closed, 13 derivation, simple, (2)184–90 open, 13 dielectric constant, (2)189–90 intrinsic mode function (IMF), (2)171 dielectric tensor, (2)279–80, (2)282–3 intrinsic permutation symmetry, (2)356, (2)400–1 electric susceptibility, (2)187–90

850 Subject index

Kramers–Kronig relations (cont.) Levin sequence transformations, 74–7 ellipticity function, (2)320–3, (2)345–6 Levin’s t transformation, 77 energy loss function, (2)270 Levin’s u transformation, 77, 718 Fourier transform approach, (2)197 light scattering, (2)239 harmonic generation susceptibility, light-cone, (2)207 (2)374–84 limit point, 39 history, (2)182, (2)207, (2)239 linear birefringence, (2)306 magnetic optical activity, (2)319–23 linear electric susceptibility tensor, (2)351 magnetic susceptibility, (2)273 asymptotic behavior, (2)372–4 magnetoreflection, (2)327–30 spatial and relativistic effects, (2)374 magneto-rotatory dispersion, (2)320–3 linear operators, 41 nonlinear electric susceptibility, (2)377–84 linear system, 638, (2)73–8 experimental verification, (2)384–6 homogeneity property, (2)74 nonlinear refractive index, (2)402 scaling property, (2)74 normal incident reflectance, (2)257–63 linear transformation, 38 optical activity, (2)345–6 linear vector space, 40 permeability, (2)273 linearity, 145 phase, (2)257–63, (2)411 Liouville–von Neumann equation, (2)364 reflectance, (2)257–63 Liouville’s theorem, 34, 70 refractive index, (2)206–7 Lipschitz condition, 12 surface impedance, (2)277–8 for Hf , 98 Kronecker delta, 18 for Hf , 316 Kummer’s confluent hypergeometric locally integrable distribution, 69 function, 257, (2)455 Loomis’s lemma, 234, 236–9, 385 Kuhn sum rule, (2)340 Loomis’s theorem, 239–41 Kutta condition, 581 Lorentz dispersion formula, (2)239 Lorentz force, (2)313–14 Lacey–Thiele theorem, (2)59 Lorentz formula (light scattering), (2)239 Laguerre polynomial, 439, (2)456 Lorentz local field, (2)338 − Hilbert transform (with weight H(x) e x), Lorentz model, (2)186, (2)223, (2)239, (2)244 451–2 Lorentz–Lorenz equation, (2)339 recurrence formula, 450 Lorentzian profile, 4, 704–5, (2)138, (2)435–6 Laguerre’s differential equation, 449 Hilbert transform, 705 Larmor frequency, (2)273 Love’s theorem, 192 Laplace transform, 267 low-pass filter, (2)120 Laplacian operator , (2)29 Lusin–Privalov theorem, 326 Laurent expansion, 31–3, 646, (2)291 Lusin’s conjecture, 325–8 Lebesgue density theorem, 229 Lebesgue dominated convergence theorem, 55, Maclaurin-type formulas, 688–90 126, 353 magnetic circular dichroism, (2)306–7, (2)310, Lebesgue integral, 52–3 (2)319–23 Lebesgue integration, 45 magnetic induction, (2)184, (2)313, (2)325 Lebesgue measure, 64 magnetic optical activity, (2)313, (2)323 Lebesgue outer measure, 49 dispersion relations, (2)319–23 Legendre function of the second kind, 439, magnetic optical rotation, (2)309 563, (2)456 magnetic permeability, (2)184 Legendre polynomial, 438–9, 693–4, 697, (2)456 magnetic resonance (NMR), (2)433–5 finite Hilbert transform, 563–7 magnetic rotation spectra, (2)309 generating function, 439 magnetic susceptibility, (2)272–4, (2)433–4 Hilbert transform, 441–6 asymptotic behavior, (2)274 orthogonality condition, 442 Kramers–Kronig relations, (2)273, (2)433 recurrence formula, 443 magnetization, (2)271, (2)335–6, (2)343 shifted, 700 magneto-optical activity, (2)313, (2)319 Legendre’s differential equation, 438 sum rules, (2)323–4 lemma, examples of magneto-optical rotation, (2)309, (2)320–23, covering, 241, 382 (2)349 Jordan, 30 magnetoreflection, (2)325–30 Loomis, 233–4, 236–7, 239, 385 dispersion relations, (2)327–30 Riemann–Lebesgue, 61–3 sum rules, (2)329 Lerch function, (2)457 magneto-rotatory dispersion, (2)320 Levi-Civita pseudotensor, (2)337 dispersion relations, (2)321–3

Subject index 851

Marcinkiewicz interpolation theorem, 365, orthogonality property, (2)17 372–3, 404 Parseval-type formula, (2)16–7 marginal spectrum, (2)178 periodic function, (2)18–21 Mathematica, 444, 704, 717, 723, (2)176, product of one-dimensional operators, (2)8 (2)181 translation operators, (2)14 mask, (2)168 translation-invariant, (2)14 maximal function, 368–73 natural optical activity, (2)307, (2)330–48 maximal Hilbert transform function, asymptotic behavior, (2)338, (2)340–2 373–86 Neumann’s formula, 563–4 directional, (2)61 Neumann’s function, (2)457 ◦ ergodic, (2)65 90 phase shift filter, 6, (2)121 n-dimensional, (2)10 nonlinear dielectric permittivity, (2)398 maximum modulus, 36 sum rule, (2)402 Maxwell equations, (2)201, (2)216, (2)275, nonlinear electric susceptibility, (2)351 (2)280, (2)332 asymptotic behavior, (2)361, (2)374–6 McLean–Elliott theorem, 226–31 crossing symmetry, (2)356, (2)383 mean ergodic theorem, (2)64 dispersion relations (one-variable), measure, 45, 48, (2)64 (2)377–84 Borel, 55 n-dimensional dispersion relations, Lebesgue, 45, 51 (2)387–8 Lebesgue outer, 49 spatial symmetry, (2)356 space, (2)64 sum rules, (2)392–7 Mehler–Sonine integrals, 459–60 table, (2)396–7 Mehler’s formula, (2)156 units, (2)352 Mellin transform, 269 nonlinear refractive index, (2)395–402 of the even and odd Hilbert transforms, crossing symmetry, (2)401 274–5 dispersion relations, (2)402 of the Hilbert transform, 269–73 meromorphic behavior, (2)398 meromorphic function, 30, 548 sum rule, (2)402 method of rotations, (2)22–3 non-tangential limit, 94 Mexican-hat wavelet, 447 non-tangential maximal function, 385 Miller indices, (2)417 norm, 18, 41, 54 Minkowski’s inequality, 205–6 weak-type (1,1), 338 Minkowski’s integral inequality, (2)23 normal dispersion, (2)182 Mittag–Leffler expansion, 35 normal incident reflectance, mixed Hilbert transform, 661 dispersion relations (2)257–63 modified Bessel function of the first kind, normalization condition for the disc, (2)48 465–9, (2)455 normalized auto-correlation function, (2)126 modified Bessel function of the second kind, normed space, 41 465–9, (2)456 nuclear magnetic resonance (NMR), 1, (2)433–5 modified Struve function, (2)456 Fourier transform, (2)435 modulus, 32, (2)412, (2)418, (2)442–3 causality issues, (2)434 modulus of elasticity, (2)448 two-dimensional, (2)435 dispersion relations, (2)448 numerical evaluation of Hilbert transforms, moment formula for the Hilbert transform, 160–4, 684–91, 701–6 594, (2)213 finite Hilbert transform, 685, 727–30 finite Hilbert transform, 530, 554–5, numerical integration, 684 603 numerical quadrature, 684 Morera’s theorem, 30 of Fourier transform, 709–11 multiple-valued function, 25 multiplicity of a zero, 546–7 Gauss–Legendre quadrature, 694, 697, 728 multiplier, 158 Gaussian, 691–701 specialized, 695, 701–6 n-dimensional Fourier transform, (2)12, Kramers–Kronig relations, 708–9 (2)25–30 Maclaurin-type formulas, 688–90 n-dimensional Hilbert transforms, (2)5–18 Simpson’s rule, 691 derivative, (2)11–2 trapezoidal rule, 690 dilation operators, (2)15 distributions, (2)32–8 odd Hilbert transform operator, 147, 258, 274 eigenfunctions, (2)17–8 on the disc, 303–4 eigenvalues, (2)17–8 one-sided Hilbert transform, 420, 609–11 Fourier transform, (2)12–4 asymptotic expansion, 422–34

852 Subject index

eigenfunctions, 612 electrodynamic description, (2)330–4 eigenvalues, 612 sum rules, (2)346–8 Fourier transform approach to inversion, 612–4 optical birefringence, (2)306 inhomogeneous singular integral equation optical constants, (2)182 involving, 614 optical convention, (2)309 inversion formula, 609–11, 614 optical rotatory dispersion, (2)306–7, (2)334–5, one-sided Z transform, 644 (2)339, (2)341, (2)345 one-to-one mapping, 37, 40, 44, 49 asymptotic behavior, (2)338, (2)340–1 onto, 44 optical theorem, (2)238 open cover, 49 optically active medium, (2)307 open interval, 13 order open set, 39 of Hölder condition, 13 operators, types of of a zero, 546 adjoint, 42 order symbols O()and o(), 11–2 analytic, (2)127 orthogonal function, 18 bounded, 157–8, 160, 191, 204, 365, orthogonal polynomials, 438–9, 452 (2)59, (2)61–3, (2)65–7 orthogonality property Calderón–Zygmund, (2)2 of finite Hilbert transform, 562 truncated, (2)3 of Hilbert transform, 174–5, 310–1, (2)232 compact, 45 of n-dimensional Hilbert transform, (2)17 conjugate Poisson (upper half plane), 375 orthonormal function, 18 conjugation, 130 oscillator strength, (2)186 continuous, 204 sum rule, (2)186 density, (2)362 Paley–Wiener log-integral theorem, (2)95–102 difference, 74 parity property, 146 dilation, 155–6, (2)14–6 Parseval formula, 21 Hermitian, 42, (2)340 Parseval-type formula, Hilbert transform operator, see also under for discrete Fourier transform, 641 Hilbert transform for discrete Hilbert transform, 671, 674 bilinear, (2)58–9 for finite Hilbert transform, 559–61 bounded, 157–8, 160, 191, 204, (2)45–7, for Hilbert transform, 170, (2)232, (2)348 (2)59, (2)61–3, (2)65–7 for Hilbert transform of distributions, 490–1 commutator relations, 156, (2)8 for Hilbert transform on disc, 305–7 continuous, 204 for n-dimensional Hilbert transform, (2)16–7 eigenfunctions, 195, (2)17–8 passive system, (2)73, (2)87 eigenvalues, 195, (2)17–8 Peierls equation, 619–20, 633 even, 147, 303–4 Peierls–Nabarro equation, 619–20 inverse, 149 pentagram symbol, (2)124 iteration property, 148 period of periodic function, 15 linearity, 145 periodic distributions, 508–15 n-dimensional, (2)5–8 Hilbert transform, 515–6 odd, 147, 303–4 periodic function, 14–5, 91, 105 operator definition, 4 Hilbert transform, 114–8, 132–4 symbol of, 258 n-dimensional Hilbert transform, (2)18–21 truncated, 83, 97–8, (2)2, (2)24 periodicity property for DFT, 640 unitary property, 174 permeability, (2)184, (2)202–3, (2)239 homothetic, 156 crossing symmetry condition, (2)272 inverse Hilbert transform, 149 Kramers–Kronig relations, (2)273 isometric, 43 permittivity, (2)183, (2)185–6, (2)189–201, linear, 41 (2)222–7, (2)231–6, (2)240–1, (2)267, Poisson (upper half plane), 375 (2)278–9, (2)298 projection, 199, 312–13, (2)127 crossing symmetry, (2)193, (2)279 reflection, 156, 160 relative, (2)183 Riesz transform, (2)25–32 tensor, (2)278–9 self-adjoint, 42 permutation symmetry, (2)356, (2)370, (2)400 sublinear, 372 phase, (2)252, (2)411–8 translation, 155–6, 160, (2)14 asymptotic behavior, (2)260–1 translation-invariant, 156, (2)14 dispersion relations, (2)257–63, (2)380–3 unitary, 43 instantaneous, (2)132, (2)177 optical activity, (2)307, (2)330–48 sum rules, (2)263–6 dispersion relations (2)345–6 table, (2)267

Subject index 853

phase problem, crystallography, (2)418 quadrature, 684 phase retrieval problem, (2)411–7 quadrature filter, 6, (2)121 phase velocity, (2)202, (2)444, (2)446 quadrature function, (2)126 Phragmén–Lindelöf theorem, 100, (2)194, quantum Liouville equation, (2)364 (2)197, (2)204, (2)244, (2)278 quantum scattering, (2)105–10 piecewise continuous function, 15 planar wing approximation, 579 radial limit, 92, 94 Plancherel formula, 21 radiative reaction force, (2)93, (2)190 plasma dispersion function, 167 Radon transform, 252, 277–85 plasma frequency, (2)185 Raman spectroscopy, (2)449 Plemelj formulas, 111–2, 545, 606 Rayleigh formulas, 462 Plemelj–Sokhotsky relations, 111, (2)41 reactance, (2)438, (2)440–2 Pochhammer symbol, 257, (2)454 reactance integral theorem, (2)442 Poincaré–Bertrand formula, 59, see also reciprocal relations, 148, 303 Hardy–Poincaré–Bertrand formula rectangular step function, 442–3, (2)143 pointwise convergence, 54 Hilbert transform, (2)146 Poisson kernel, 88, 142, 375, 379–80 Redheffer extension, (2)44–7 for the disc, 91–4, 296, 339 reduced Hilbert transform, 609 Poisson integral reflectance, (2)252 for the disc, 89–91 dispersion relations (2)257–63 for the half plane, 85–9 sum rules, (2)263–6 Poisson integral formula, 87 table, (2)267 Poisson operator, upper half plane, 375 reflection formula (gamma function), 469 polarizability, (2)184, (2)202, (2)338–9 reflection operator, 156, 304–5 polarization vectors, (2)184, (2)308 reflectivity, (2)253, (2)257–67 pole of order m, 25 amplitude, (2)253 simple, 25 asymptotic behavior, (2)255, (2)260 polygamma function, (2)457 circularly polarized modes, (2)326 polylogarithm function, (2)457 crossing symmetry, (2)327 polynomial, types of complex, (2)252–63 Chebyshev, of the first kind, 439, 452, 456, crossing symmetry relation, (2)255, (2)457 (2)257 Chebyshev, of the second kind, 439, 452, (2)457 refractive index, (2)183, (2)201–8 Gegenbauer, 439, 452, 472, 570, (2)455 analytic properties, (2)203–4 Hermite, 439, 446–9, (2)155–7, (2)163, (2)455 asymptotic behavior, (2)204 Jacobi, 439, 452, 472, (2)456 N+ and N−, (2)309–19 Laguerre, 439, 449–52, (2)456 complex, (2)202 Legendre, 438–46, 563–7, 693–4, 697, 700, crossing symmetry relation, (2)206 731, (2)456 Kramers–Kronig relations, (2)206–7 positron–atom scattering, (2)429 single subtracted dispersion relation, (2)211 potential scattering, (2)425–8 sum rules, (2)227–31, (2)236–9, (2)299 dispersion relations, (2)426–8 table, (2)242–3 pre-envelope, (2)126 regular distribution, 69 principal axis system, (2)279, (2)309 regular function, 25 principal part, 32 relative distributional inequality, 381 principal value, see Cauchy principal value relative magnetic permeability, (2)272 principle of relativistic causality, (2)207 relative multiplier, (2)130 principle of superposition, (2)74 relative permittivity, (2)183 principle of the argument, 548 removable singularity, 25 Privalov, problem of, 544 renormalized Hilbert transform, (2)47 Privalov’s theorem, 316–18 residue, 33 probability current density, (2)106 resistance integral theorem, (2)442 probability density, (2)106 resistive-shunted junction model, (2)407 projection operators, 199, 312–3 resonance condition, (2)78 proper subset, 39 response function, 638, (2)73, (2)80, (2)87, (2)90, pseudofunction, 476 (2)111, (2)191, (2)194, (2)206, (2)224, pseudoscalar, (2)331 (2)254, (2)280–1, (2)336, (2)353, (2)355, psi (digamma) function, 178, (2)457 (2)401 p.v.(1/x), 69, 476, 484 complex, (2)87–9 Fourier transform, 489 reverse Hölder inequality, 389–90 Hilbert transform, 491 Richardson extrapolation, 71–4

854 Subject index

Riemann, problem of, 543 sectionally analytic function, 110 Riemann integral, 45–6, 50–2 sectionally continuous function, 110 Riemann integration, 45 self-adjoint operator, 42 Riemann mapping theorem, 37 Sellmeier’s dispersion formula, (2)248 Riemann surface, 26 semi-discrete Hilbert transform, 661 Riemann zeta function, 70, 662 semi-infinite Hilbert transform, 609 Riemann–Hilbert problem, 544–6 semicircular contour, 31, 86 Riemann–Lebesgue lemma, 61–3 semigroup, 40, (2)60, (2)149, (2)161–2, (2)181 Riesz transform, (2)25–32 sequential system, (2)79 bounded property, (2)29 series expansion approach for the Hilbert Fourier transform, (2)25 transform, 177 Riesz’s inequality, 203–11, 331–4, (2)21, (2)38, series summation, 70 (2)234, (2)236 set, types and properties for discrete sequence, 661 accumulation point, 39 on the circle, 318–23 Borel, 55 truncated Hilbert transform, 254 bounded, 39 ring, (2)64 closed, 39 σ -ring, (2)64 compact, 45 Rodrigues’ formula, 449, (2)156 complement, 39 Rosenfeld formula, (2)339 difference, 39 asymptotic behavior, (2)340–2 empty, 39 rotated ergodic Hilbert transform, (2)23 intersection, 39 rotational strength, (2)339 limit point, 39 rotations, method of, (2)22–3 subset, 39 open, 39, 44, 49 S symmetry element, (2)330–1 n proper subset, 39 S-matrix, (2)106 union, 39 sampling theorem, see Shannon sampling theorem, see Whittaker–Shannon–Kotel’nikov theorem Whittaker–Shannon–Kotel’nikov theorem saturation, (2)392, (2)433 shift-invariant system, 645, (2)80 scale changes, 150–5, 304–5 shifted Legendre polynomial, 700 scattering amplitude, light, (2)239, (2)244–7 sifting property, 65 scattering factor, (2)417 sign function, see signum function scattering theory, (2)102 Born approximation, (2)424–5 signum function (sgn), 6, (2)142, (2)146, (2)456 causality conditions, (2)104 cross-section, (2)105 Fourier transform, 489–90 elastic scattering amplitude, (2)425–8 signal processing, (2)119 electron–atom, (2)422, (2)428–32 analog signals, (2)119 potential scattering, (2)425–8 analytic signal, 181, (2)126–35, (2)137–8, quantum scattering, (2)105–10 (2)162, (2)169, (2)176 s-wave scattering, (2)103, (2)105, (2)118 discrete signals, 637, 644, 649–52, 656–60, S-function, (2)103, (2)106, (2)108 (2)119 crossing condition, (2)109 digital signals, (2)119 S-matrix, (2)106 energy, (2)120, (2)125, (2)162 scattered wave, (2)103, (2)106 filter, (2)120 spherically symmetric scatterer, (2)102 band-pass, (2)120 Schrödinger equation, (2)363, (2)423 band-stop, (2)120 time-dependent, (2)105, (2)364 high-pass, (2)120 Schützer–Tiomno requirement, (2)108 Hilbert, 650, (2)121 Schwartz distributions, 68 Hilbert transformer, (2)121 Schwartz test functions, 69 low-pass, (2)120 Schwartz’s theorem, (2)111, (2)113 90-degree phase shift, (2)121 Schwartz–Sobolev distributions, 70 quadrature, (2)121 Schwarz inequality, see Fourier transform, (2)120 Cauchy–Schwarz–Buniakowski inequality Hilbert transfer function, (2)121 second-harmonic generation (SHG), (2)357, pre-envelope, (2)126 (2)393 simple function, 52 second-order nonlinear electric susceptibilities, simple pole, 25 (2)351–2, (2)355, (2)357–8, (2)369, (2)371, simply connected region, 28 (2)386 Simpson’s rule, 691

Subject index 855

sinc function, 185, 719, (2)145, (2)419, (2)457 spatial dispersion, (2)280–90 Hilbert transform, (2)147 spatial-dependent electric displacement, (2)281 ( ) sine integral function, si x , 470, (2)457 specialized Gaussian quadrature, 695, 701–6, ( ) sine integral function, Si x , 431, 470, 522, (2)457 708–9 sine-exponential integral, (2)457 application to the Hilbert transform, 701–6 sine-Gordon equation, 620 spectrum, (2)131, (2)135 sine-Hilbert equation, 620–4 generalized, 624 spherical Bessel functions, 462, (2)455, (2)457 singular distribution, 69 differential equation, 462 singular integral equation, 2, 59, 588 Hilbert transform, 462–4 Fourier transform approach, 599–601 recurrence formula, 463 involving distributions, 630–1 spin–lattice relaxation time, (2)273, (2)434 singular integro-differential equations, 593 spin–spin relaxation time, (2)433 singular points, 25 square wave, 289–91 singularity, essential, 25 stable system, 649, (2)80, (2)117 removable, 25 stationary system, (2)79–80 skew-symmetric character, 147 stationary time series, (2)170 slow growth distribution, 481 Stein–Weiss theorem, 241–6 Sobolev space, 482 step function, 50–1, (2)139 Sokhotsky–Plemelj formulas, 111, (2)41, (2)413 Heaviside, 450, (2)139, (2)141–2 solitary wave, 624–6 rectangular, 443, (2)143 soliton, 623, 625 space, function or distribution Stieltjes transform, 265–7, 335, 609 Ck , 69 asymptotic expansion, 420–2 Ck , 69 stiffness, (2)447 0∞ C , 69 strain, (2)447 ∞ stress, (2)447 C0 , 69 D(R), 69, 478 strong-type estimates, 203, 364, 403 D(Rn), 69 Struve’s function, 460, (2)455 D p , 479 Hilbert transform, 460–2 L D , 479 modified, 465–6, (2)456 D q , 479, (2)33, (2)35–8 sublinear operator, 372 L D+, 480 subset, 39 E, 481 subtracted dispersion relation, 685, (2)211, (2)384 E , 481 sum frequency generation, (2)358 H , 516–7, (2)36 sum rule, (2)186, (2)198, (2)203, (2)219, (2)222, H , 516–7, (2)36 (2)227, (2)231, (2)236 O C , 482 conductance, (2)268, (2)289–90 Pτ , 482 dielectric constant, (2)198, (2)222–7, (2)402 P τ , 482 table, (2)240–1 S , 480–1, (2)35 dielectric tensor, (2)284–5, (2)287–290 S , 481 energy loss function, (2)270–1 S 1, 518, (2)35 Z, 481 f , (2)225, (2)228, (2)230, (2)246, (2)250, (2)268, (2)270, (2)299, (2)301 Z , 481 oscillator strength, (2)186 Z1, 518, (2)35 Z Kuhn, (2)340 1, 518 Euclidean, 42, 49 magneto-optical activity, (2)323–4 Hardy, 43 magnetoreflection, (2)325, (2)329 Hilbert, 42 nonlinear dielectric permittivity, (2)402 measure, (2)64 nonlinear refractive index, (2)402 normed, 41 nonlinear susceptibilities, (2)392–5 periodic distributions, 482, 508 table, (2)396–7 Sobolev, 482 test functions of rapid decay, 480 optical activity, (2)346–8 topological, 44 phase, (2)263–6 ultradistributions, 481, 516 table (2)267 unitary functions, 508–9 reactance, (2)442

856 Subject index

sum rule (cont.) cosine form, (2)525–6 refractive index, (2)227–31, (2)299, cotangent form, (2)528–32 (2)402 Hilbert transforms He and Ho, (2)533 table, (2)242–3 one-sided Hilbert transform, (2)526–7 reflectance, (2)263–6 multiple Hilbert transforms, (2)518–9 table, (2)267 powers and algebraic functions, (2)459–64 resistance, (2)442 pulse and waveforms, (2)508–16 Thomas–Reiche–Kuhn, (2)225, (2)284, special functions (2)324 Anger functions, (2)507 weight functions, (2)236–9 Bessel functions of the first kind of integer summable, square, 46 order, (2)499 summable function, 66 Bessel functions of the first kind of fractional summation convention, 161, (2)457 order, (2)500–2 summation rearrangement, 440, 445, Bessel functions of the second kind of 449, 451 fractional order, (2)502 superconvergence relations, (2)249 cosine integral function, (2)506 superconvergence theorem, (2)222 Hermite polynomials, (2)497–8 superhilbert transform, (2)70 Laguerre polynomials, (2)498–9 support, 68 Legendre polynomials, (2)494–6 supremum, 36 miscellaneous, (2)507 surface acoustic impedance, (2)447 modified Bessel functions of the first kind, surface impedance, (2)274–8 (2)504 asymptotic behavior, (2)278 modified Bessel functions of the second kind, crossing symmetry relation, (2)277 (2)504 dispersion relations, (2)277–8 product of Bessel functions of the first kind surjective function, 44 of fractional order, (2)503 susceptance (circuit), (2)440 sine integral function, (2)506 system, (2)73 spherical Bessel functions of the first kind, active, (2)73 (2)505 causal, (2)81 spherical Bessel functions of the second kind, causal condition, 649 (2)505 homogeneity property, (2)74 Struve functions, (2)507 linear, 638, (2)73–4 trigonometric functions, (2)471–91 passive, (2)73 Taylor’s theorem, (2)112 scaling property, (2)74 Tchebichef polynomial, see Chebyshev sequential systems, (2)79 polynomial shift-invariant, 645, (2)80 temperate distribution, 481 stable, 645, (2)80 tempered distribution, 481 stationary systems, (2)79–80 Hilbert transform, 517 time-invariant, 645 Temple’s theory, 65 transfer function, (2)82 tensor product of distributions, 491–5, (2)32, (2)35 symbol of H, 258 test function, 68 symmetry operation, theorems, names of Cn, (2)310 Akhiezer’s, 200–1 inversion element, (2)330 Bedrosian’s, 184–7, 455–7, (2)131 plane of symmetry, (2)330 Boas’, (2)49, (2)54 Sn, (2)330 Bochner’s, (2)83 Cauchy–Goursat, 29 table of Hilbert transforms, Cauchy integral, 29 distributions, (2)517 Cauchy residue, 33 exponential functions, (2)465 central slice, 281 finite Hilbert transforms, closure, (2)340 interval [−1, 1], (2)519–22 Fatou’s, 94, 320, 326 interval [0, 1], (2)523–4 Fubini’s, 55 interval [a, b], (2)525 Green’s, 331, 341 general properties, (2)458 Helson–Szegö, 386–8 hyperbolic functions, (2)466–70 Hilbert transform product, 184–7 inverse trigonometric and inverse hyperbolic Hunt–Muckenhoupt–Wheeden, 395–9, 679–80 functions, (2)493 Kober’s, 170, 212 logarithmic functions, (2)492 Kolmogorov’s, 338, 364, 367 miscellaneous cases Kolmogorov’s weak-type, 364

Subject index 857

Lacey–Thiele, (2)59 bilinear Hilbert, (2)58–9 Lebesgue density, 229 Boas, (2)49–57 Lebesgue dominated convergence, 55, Boas–Heinig, (2)57 126, 353 Cauchy, 85 Liouville’s, 34 causal, 215, (2)104 Loomis’, 239–41 discrete Fourier, 637–44, 651, 673, (2)168–9 Love’s, 192 discrete fractional Fourier, (2)163–8 Lusin–Privalov, 326 discrete fractional Hilbert, (2)168–9 Marcinkiewicz interpolation, 365, 372–3, 404 discrete Hilbert, 637, 660–75, 678–80, (2)168 McLean–Elliott, 226–31 finite Hilbert, 443, 452, 521–82, 601–9, 685, mean ergodic, (2)65 727–30 Morera’s, 30 Fourier, 1, 19–21, 158, 252–8, 276 optical, (2)245 fractional Fourier, (2)149–59 Paley–Wiener log-integral, (2)95–102 fractional Hilbert, (2)159–63, (2)169 Phragmén–Lindelöf, 100, (2)194, (2)197, Hartley, 262 (2)204, (2)244, (2)278 Hilbert, see under Hilbert transform Privalov’s, 316 Hilbert–Huang, (2)170–8 reactance integral, (2)442 Hilbert–Stieltjes, 231–41, 364–65 resistance integral, (2)442 inverse Mellin, 270 Riemann mapping, 37 Laplace, 267 Schwartz’s, (2)111, (2)113 Mellin, 269 Stein–Weiss, 241–6 one-sided Hilbert, 420, 422–34, 609–14 superconvergence, (2)222 Radon, 277–85 Taylor’s, (2)112 Riesz, (2)25–32 Titchmarsh’s, 215–23 Stieltjes, 265–7, 335 distributions, (2)110 Tricomi, 522 Tonelli’s, 55–7 two-sided Laplace, 267 Tricomi’s, 187–95, 313–5, 529–36, 602–3 unilateral Laplace, 267 Viète–Girard, 235, 248 Wiener–Lee, 6, 106 Vitali’s covering, 241 Z, 644–8 Whittaker–Shannon–Kotel’nikov, 344, transformation (2)419–20 bilinear, 38 Young’s, 378, (2)53 linear, 38 Zygmund’s L log L, 340–3 translation operator, 155, (2)14 thin airfoil problem, 577–82 translation-invariant operator, 156 third-harmonic generation (THG), (2)357–8, transverse relaxation time, (2)273, (2)433 (2)376, (2)385 trapezoidal rule, 690 third-order nonlinear electric susceptibilities, triangle inequality, 41 (2)351, (2)356–8, (2)389, (2)398–401 triangular function, 186, (2)145 Hilbert transform, (2)147 Thomas–Reiche–Kuhn sum rule, (2)225, (2)284 Tricomi transform, 522 time-dependent Schrödinger equation, (2)105, Tricomi’s identity, 529, 543, 560, 563, 602–3, 609, (2)364 615, (2)234 time-invariance, (2)80, (2)111 Tricomi’s theorem, 187 time-invariant system, 649 truncated Calderón–Zygmund operator, (2)3 time series, (2)170 truncated Hilbert transform, 83, 226, 253, 373, stationary, (2)170 375, 378, 521, (2)2–3, (2)24 Titchmarsh formulas of the Parseval-type, 170–1 cotangent form, 373, 527–9 Titchmarsh’s theorem, 215, 215–23, (2)77, (2)86, two-dimensional Fourier transform, 280–1 (2)90–3, (2)102, (2)104, (2)110–5, (2)205, two-sided Laplace transform, 267 (2)336, (2)343, (2)401 two-sided Z transform, 644 distributions, (2)110 type (p, q), 364 Tonelli’s theorem, 57 topological space, 44 ultradistributions, 481, 483 topology, 44 Hilbert transform, 516 total scattering cross-section, (2)245 ultraspherical function of the second kind, 570 total variation, 232 ultraspherical polynomial (Gegenbauer trace, (2)363 polynomial), 439, 452, 472, 570, (2)455 transfer function, Hilbert, (2)121, (2)148–9, uniform convergence, 47 (2)168 unilateral Laplace transform, 267 transform, types of union of sets, 39 bilateral Laplace, 267 unit impulse function, 63

858 Subject index

unit rectangular step function, 442–3 Weierstrass M-test, 47 unit sample sequence, 653 weighted Hilbert transforms, 354–3, 395–8 unit step sequence, 652 weighted inequalities Fourier transform, 657 discrete Hilbert transform, 679–80 unit triangular function, 186, (2)145 for He and Ho, 399–402 Hilbert transform, (2)147 for He and Ho, 401 unitary function, 509–16 with two weights, 403–8 unitary operator, 43 with vanishing moments, 402–3 unitary property of Hilbert transform operator, weighted norm inequalities, 354 174, (2)234 weighted sum rules, (2)236–9 Whittaker’s cardinal series, see Vakman conditions, (2)132, (2)134 Whittaker–Shannon–Kotel’nikov theorem Van Kampen causality condition, (2)109 Whittaker–Shannon–Kotel’nikov theorem, 344, Vandermonde determinant, 72–3 (2)419–20 vector space (linear), 40 Wiener–Lee transform, 6, 106 vectorial Hilbert transform, (2)60 velocity X-ray crystallography, (2)417–22 front, (2)207 group, (2)228 Young’s form, 84, 98, 127, 130, 135 phase, (2)202, (2)444 Young’s inequality, 378, (2)53 Verdet constant, (2)310 Young’s theorem, 378 Viète–Girard theorem, 235, 248 viscoelastic behavior, (2)447–8 Z transform, 644 Vitali’s covering theorem, 241 Volterra integral equations, 589 of a causal sequence, 652 one-sided, 644 wave number, (2)202 of a product, 647 complex, (2)444 two-sided, 644 waveform analysis, 439, 446 Zayed’s definition fractional analytical signal, weak derivatives, 482 (2)169 weak-type inequalities, 364 Zayed’s definition fractional Hilbert transform, (1,1), 338, 364 (2)169 (1,1) norm, 338 Zygmund L log L class, 341 (p, q), 364 Zygmund-type inequalities (on the circle), 411 Weber’s functions, 453, 471, (2)455 Zygmund’s inequality, 340, (2)21 Hilbert transform, 471 weighted version, 363 Weierstrass factorization formula, 36 Zygmund’s L log L theorem, 340–3