Bibliography

[1] E. Abe, Hopf algebras. Cambridge University Press, 1980. [2] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions. United States Department of Commerce, 1965. [3] M.S. Agranovich, Spectral properties of elliptic pseudodifferential operators on a closed curve. (Russian) Funktsional. Anal. i Prilozhen. 13 (1979), no. 4, 54–56. (English translation in and Its Applications. 13, p. 279–281.) [4] M.S. Agranovich, Elliptic pseudodifferential operators on a closed curve. (Russian) Trudy Moskov. Mat. Obshch. 47 (1984), 22–67, 246. (English translation in Transactions of Moscow Mathematical Society. 47, p. 23–74.) [5] M.S. Agranovich, Elliptic operators on closed manifolds (in Russian). Itogi Nauki i Tehniki, Ser. Sovrem. Probl. Mat. Fund. Napravl. 63 (1990), 5–129. (English translation in Encyclopaedia Math. Sci. 63 (1994), 1–130.) [6] B.A. Amosov, On the theory of pseudodifferential operators on the circle. (Russian) Uspekhi Mat. Nauk 43 (1988), 169–170; translation in Russian Math. Surveys 43 (1988), 197–198. [7] B.A. Amosov, Approximate solution of elliptic pseudodifferential equations on a smooth closed curve. (Russian) Z. Anal. Anwendungen 9 (1990), 545– 563. [8] P. Antosik, J. Mikusi´nski and R. Sikorski, Theory of distributions. The se- quential approach. Warszawa. PWN – Polish Scientific Publishers, 1973. [9] A. Baker, Matrix Groups. An Introduction to Lie Group Theory. Springer- Verlag, 2002. [10] J. Barros-Neto, An introduction to the theory of distributions . Marcel Dekker, Inc., 1973. [11] R. Beals, Advanced mathematical analysis. Springer-Verlag, 1973. [12] R. Beals, Characterization of pseudodifferential operators and applications. Duke Mathematical Journal. 44 (1977), 45–57. 684 Bibliography

[13] A.P. Bergamasco and P.L. da Silva, Solvability in the large for a class of vector fields on the torus. J. Math. Pures Appl. 86 (2006), 427–447. [14] J. Bergh and J. L¨ofstr¨om, Interpolation spaces. An introduction. Springer- Verlag, 1976. [15] G. Boole, Finite differences. 4th edition, Library of Congress catalogue card no. 57-8495. (1st edition 1860). [16] J. Bourgain, Exponential sums and nonlinear Schr¨odinger equations. Geom. Funct. Anal. 3 (1993), 157–178. [17] J. Bourgain, Fourier transform restriction phenomena for certain lattice sub- sets and applications to nonlinear evolution equations. I. Schr¨odinger equa- tions. Geom. Funct. Anal. 3 (1993), 107–156. [18] J. Bourgain, Global solutions of nonlinear Schr¨odinger equations. American Mathematical Society Colloquium Publications, 1999. [19] G.E. Bredon: Introduction to Compact Transformation Groups. Academic Press, 1972. [20] T. Br¨ocker and T. tom Dieck, Representations of Compact Lie Groups. Springer-Verlag, 1985. [21] A.P. Calder´on, Commutators of singular integral operators. Proc. Nat. Acad. Sci. USA 53 (1965), 1092–1099. [22] A.P. Calder´on and R. Vaillancourt, On the boundedness of pseudo-differen- tial operators , J. Math. Soc. Japan 23 (1971), 374–378. [23] R.R. Coifman and Y. Meyer, Au-del`ades op´erateurs pseudo-diff´erentiels. Ast´erisque 57, Soci´et´eMath. de France. 1978. [24] L. Comtet, Advanced combinatorics. Dordrecht. D.Reidel Publishing Com- pany, 1974. [25] H.O. Cordes, On compactness of commutators of multiplications and convo- lutions, and boundedness of pseudo-differential operators, J. Funct. Anal. 18 (1975), 115–131. [26] H.O. Cordes, On pseudodifferential operators and smoothness of special Lie group representations. Manuscripta Math. 28 (1979), 51–69. [27] H.O. Cordes, The technique of pseudodifferential operators . Cambridge Uni- versity Press, 1995. [28] D. Crespin, Hahn–Banach Theorem Implies Riesz Theorem. Portugaliae Mathematica 51 (1994), 217–218. [29] G. David, Wavelets and singular integrals on curves and surfaces. Springer- Verlag, 1992. [30] J.J. Duistermaat, Fourier integral operators. Birkh¨auser, 1996. Bibliography 685

[31] J.J. Duistermaat and J.A. Kolk, Lie groups. Springer-Verlag, 2000. [32] J. Dunau, Fonctions d’un operateur elliptique sur une variete compacte. J. Math. Pures et Appl. 56 (1977), 367–391. [33] Y.V. Egorov, B.-W. Schulze, Pseudo-differential operators, singularities, ap- plications. Operator Theory: Advances and Applications, 93, Birkh¨auser, 1997. [34] J. Elschner, Singular ordinary differential operators and pseudodifferential equations. Springer-Verlag, 1985. [35] G.B. Folland. Real Analysis. Modern techniques and their applications. Sec- ond edition. A Wiley-Interscience Publication. John Wiley and Sons, Inc., 1999. [36] G.B. Folland, Harmonic analysis in phase space. Princeton Univ. Press, 1989. [37] H. Freudenthal and H. de Vries, Linear Lie Groups. Academic Press, 1969. [38] S.A. Gaal, Linear Analysis and Representation Theory. Springer-Verlag, 1973. [39] I.M. Gelfand and G.E. Shilov, Generalized functions. Vols. 1–3, Academic Press, 1968 [40] F. Geshwind and N.H. Katz, Pseudodifferential operators on SU(2), J. Fourier Anal. Appl. 3 (1997), 193–205. [41] P. Glowacki, A symbolic calculus and L2-boundedness on nilpotent Lie groups , J. Funct. Anal. 206 (2004), 233–251. [42] T. Gramchev, P. Popivanov and M. Yoshino, Global solvability and hypoel- lipticity on the torus for a class of differential operators with variable coef- ficients. Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), 53–57. [43] T. Gramchev, P. Popivanov and M. Yoshino, Global properties in spaces of generalized functions on the torus for second order differential operators with variable coefficients. Rend. Sem. Mat. Univ. Politec. Torino 51 (1993), 145–172. [44] T. Gramchev, Simultaneous normal forms of perturbations of vector fields on tori with zero order pseudodifferential operators. Symmetry and perturbation theory (Rome, 1998), 187–195, World Sci. Publ., River Edge, NJ, 1999. [45] A. Grigis and J. Sj¨ostrand, Microlocal analysis for differential operators. An introduction. Cambridge University Press, 1994. [46] P.R. Halmos, Naive Set Theory. Springer-Verlag. 1974. [47] S. Helgason, Differential geometry and symmetric spaces. Academic Press, 1962. 686 Bibliography

[48] S. Helgason, Topics in harmonic analysis on homogeneous spaces. Birk- h¨auser, 1981. [49] S. Helgason, Groups and geometric analysis , Academic Press, 1984. [50] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis I. Springer-Verlag, 1963. [51] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis II. Springer-Verlag, 1963. [52] F.B. Hildebrand, Finite-difference equations and simulations. Prentice-Hall, Inc., 1968. [53] E. Hille and R.S. Phillips, Functional Analysis and Semi-Groups. American Mathematical Society, 1981. [54] M.W. Hirsch, Differential Topology. Springer-Verlag, 1976. [55] L. H¨ormander, The analysis of linear partial differential operators III. Springer-Verlag, 1985. [56] L. H¨ormander, The Analysis of Linear Partial Differential Operators IV. Springer-Verlag, 1985. [57] R. Howe, A symbolic calculus for nilpotent groups , Operator algebras and group representations, Vol. I (Neptun, 1980), 254–277, Monogr. Stud. Math., 17, Pitman, Boston, MA, 1984. [58] T. Husain, Introduction to Topological Groups. W.B. Saunders Company, 1966. [59] V. Hutson and J.S. Pym, Applications of functional analysis and operator theory. Academic Press, 1980. [60] Ch. Jordan, Calculus of finite differences. New York. Chelsea Publishing Company, 1950. [61] Y. Katznelson, An Introduction to Harmonic Analysis. Dover, 1976. [62] O. Kelle and G. Vainikko, A fully discrete Galerkin method of integral and pseudodifferential equations on closed curves. Journal for Analysis and its Applications. 14 (1995), 593–622. [63] J.L. Kelley, I. Namioka et al.: Linear Topological Spaces. D. Van Nostrand Company, Inc. Princeton, New Jersey, 1963. [64] A.A. Kirillov, Elements of the Theory of Representations. Springer-Verlag, 1976. [65] A. Klimyk and K. Schm¨udgen, Quantum Groups and Their Representations. Springer-Verlag, 1997. [66] K. Knopp, Theory and application of infinite series. Glasgow. Blackie & Son Limited, 1948. Bibliography 687

[67] D.E. Knuth, Two notes on notation. Amer. Math. Monthly 99 (1992), 403– 422. [68] J.J. Kohn and L. Nirenberg, On the algebra of pseudo-differential operators. Comm. Pure Appl. Math. 18 (1965), 269–305. [69] R. Kress, Linear integral equations. Springer-Verlag, 1989. [70] E. Kreyszig, Introductory Functional Analysis with Applications. John Wiley & Sons 1989. [71] H. Kumano-go, Pseudodifferential operators. MIT Press, Cambridge, Mass.- London, 1981. [72] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag. 1972. [73] S. Majid, Foundations of Quantum Group Theory. Cambridge University Press, 1995. [74] S. Majid, A Quantum Groups Primer. Cambridge University Press, 2002. [75] D.K. Maslen, Efficient computation of Fourier transforms on compact groups. J. Fourier Anal. Appl. 4 (1998), 19–52. [76] W. McLean, Local and global description of periodic pseudodifferential oper- ators. Math. Nachr. 150 (1991), 151–161. [77] G.A. Meladze and M.A. Shubin, A of pseudodifferential operators on unimodular Lie groups . J. Soviet Math. 47 (1989), 2607–2638. [78] A. Melin, Parametrix constructions for right invariant differential operators on nilpotent groups , Ann. Global Anal. Geom. 1 (1983), 79–130. [79] S.T. Melo, Characterizations of pseudodifferential operators on the circle . Proc. Amer. Math. Soc. 125 (1997), 1407–1412. [80] S.T. Melo, Smooth operators for the regular representation on homogeneous spaces. Stud. Math. 142 (2000), 149–157. [81] R.B. Melrose, Geometric scattering theory. Cambridge Univ. Press, 1995. [82] S. Molahajloo and M.W. Wong, Pseudo-differential operators on S1, in “New Developments in Pseudo-Differential Operators”, Operator Theory: Advances and Applications, Vol. 189, 2009, 297–306. [83] S. Molahajloo, Pseudo-differential operators on Z, in “ Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations”, Operator Theory: Advances and Applications, to appear. [84] A. Mohammed and M.W. Wong, Sampling and pseudo-differential operators , in “New Developments in Pseudo-Differential Operators”, Operator Theory: Advances and Applications, Vol. 189, 2009, 323–332. 688 Bibliography

[85] N.E. N¨orlund, Vorlesungen ¨uber Differenzenrechnung. Berlin. Verlag von Julius Springer, 1924. [86] A. Pietsch, Nuclear locally convex spaces. Springer-Verlag, 1972. [87] D.W. Robinson, Elliptic operators and Lie groups . Oxford University Press, 1991. [88] W. Rudin, Functional Analysis. Tata McGraw-Hill, 1974. [89] W. Rudin, Real and Complex Analysis. McGraw-Hill 1987. [90] M.V. Ruzhansky, Singularities of affine fibrations in the regularity theory of Fourier integral operators , Russian Math. Surveys, 55 (2000), 93–161. [91] M. Ruzhansky, Regularity theory of Fourier integral operators with complex phases and singularities of affine fibrations. CWI Tracts, vol. 131, 2001. [92] M. Ruzhansky, Lp-distributions on symmetric spaces . Results Math. 44 (2003), 159–168. [93] M. Ruzhansky and M. Sugimoto, Global calculus of Fourier integral oper- ators, weighted estimates, and applications to global analysis of hyperbolic equations , in Advances in pseudo-differential operators, 65–78, Oper. The- ory Adv. Appl., 164, Birkh¨auser, 2006. [94] M. Ruzhansky and M. Sugimoto, Global L2 boundedness theorems for a class of Fourier integral operators, Comm. Partial Differential Equations , 31 (2006), 547–569. [95] M. Ruzhansky and M. Sugimoto, Weighted L2 estimates for a class of Fourier integral operators, arXiv:0711.2868v1 [96] M. Ruzhansky and V. Turunen, On the Fourier analysis of operators on the torus , Modern trends in pseudo-differential operators, 87-105, Oper. Theory Adv. Appl., 172, Birkh¨auser, Basel, 2007. [97] M. Ruzhansky and V. Turunen, Quantization of pseudo-differential operators on the torus , arXiv:0805.2892v1 [98] M. Ruzhansky and V. Turunen, Global quantization of pseudo-differential operators on compact Lie groups, SU (2) and S3, arXiv:0812.3961v1. [99] Yu. Safarov, Pseudodifferential operators and linear connections. Proc. Lon- don Math. Soc. 74 (1997), 379–416. [100] X. Saint Raymond, Elementary introduction to the theory of pseudodifferen- tial operators. Studies in Advanced . CRC Press, Boca Raton, FL, 1991. [101] J. Saranen and G. Vainikko, Periodic integral and pseudodifferential equa- tions with numerical approximation. Springer-Verlag, 2002. Bibliography 689

[102] J. Saranen, W.L. Wendland, The Fourier series representation of pseudod- ifferential operators on closed curves. Complex Variables Theory Appl. 8 (1987), 55–64. [103] L. Schwartz, Sur l’impossibilit´ede la multiplication des distributions , C. R. Acad. Sci. Paris 239 (1954), 847–848. [104] L. Schwartz, Th´eorie des distributions, I, II . 2nd ed., Hermann, Paris, 1957. [105] L. Schwartz, M´ethodes math´ematiques pour les sciences physiques. (French) Hermann, Paris 1961 [106] I.E. Segal, An extension of Plancherel’s formula to separable unimodular groups. Ann. Math. 52 (1950), 272–292. [107] I.E. Segal, A non-commutative extension of abstract integration. Ann. Math. 57 (1953), 401–457. [108] V.A. Sharafutdinov, Geometric symbol calculus for pseudodifferential opera- tors. I [Translation of Mat. Tr. 7 (2004), 159–206]. Siberian Adv. Math. 15 (2005), 81–125. [109] T. Sherman, Fourier analysis on the sphere , Trans. Amer. Math. Soc. 209 (1975), 1–31. [110] T. Sherman, The Helgason Fourier transform for compact Riemannian sym- metric spaces of rank one , Acta Math. 164 (1990), 73–144. [111] M.A. Shubin, Pseudodifferential operators and spectral theory. Springer- Verlag, 1987. [112] J. Sj¨ostrand, Microlocal analysis. In Development of mathematics 1950– 2000 , 967–991, Birkh¨auser, Basel, 2000. [113] R.M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable. Ann. of Math. 92 (1970), 1–56. [114] M. Spivak, A comprehensive introduction to differential geometry. Publish or Perish, Inc., Boston, Mass., 1975. [115] L.A. Steen and J.A. Seebach, Jr.: Counterexamples in Topology. Dover Pub- lications, Inc., 1995. [116] J.F. Steffensen, Interpolation. New York. Chelsea Publishing Company, 1950. [117] E.M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, 1993. [118] H. Stetkaer, Invariant pseudo-differential operators. Math. Scand. 28 (1971), 105–123. [119] W.F. Stinespring, Integration theorems for gages and duality for unimodular groups. Trans. Amer. Math. Soc. 90 (1959), 15–56. 690 Bibliography

[120] R. Strichartz, Invariant pseudo-differential operators on a Lie group. Ann. Scuola . Sup. Pisa 26 (1972), 587–611. [121] R. Strichartz, Local harmonic analysis on spheres. J. Funct. Anal. 77 (1988), 403–433. [122] M. Sugiura, Unitary Representations and Harmonic Analysis – an Introduc- tion. Kodansha Ltd., 1975. [123] P. Suppes, Axiomatic Set Theory. Dover Publications, Inc., 1972. [124] S.L. Svensson, Pseudodifferential operators – a new approach to the boundary problems of physical geodesy. Manuscripta Geodaetica 8 (1983), 1–40. [125] M.E. Sweedler, Hopf Algebras. W.A. Benjamin, Inc. 1969. [126] K. Tapp, Matrix Groups for Undergraduates. American Mathematical Soci- ety, 2005. [127] M.E. Taylor, Noncommutative microlocal analysis. Memoirs AMS 52 (1984), No. 313. [128] M.E. Taylor, Noncommutative harmonic analysis. Mathematical Surveys and Monographs, Vol. 22, Amer. Math. Soc., 1986. [129] M.E. Taylor, Pseudodifferential operators . Princeton University Press, 1981. [130] M.E. Taylor, Pseudodifferential operators and nonlinear PDE. Birkh¨auser, 1991. [131] M.E. Taylor, Partial differential equations. Vol. III. Nonlinear equations. Springer-Verlag, 1997. [132] M.E. Taylor, Beals–Cordes-type characterizations of pseudodifferential oper- ators. Proc. Amer. Math. Soc. 125 (1997), 1711–1716. [133] F. Treves: Topological Vector Spaces, Distributions and Kernels. Adademic Press. New York, 1967. [134] F. Treves, Introduction to pseudodifferential and Fourier integral operators. Plenum Press. Vol. 1, Vol. 2., 1980. [135] V. Turunen, Commutator characterization of periodic pseudodifferential op- erators. Z. Anal. Anw. 19 (2000), 95–108. [136] V. Turunen, Periodic convolution integral operators. An undergraduate essay at Helsinki University of Technology. 19 p., 1996. [137] V. Turunen, Symbol analysis of periodic pseudodifferential operators. Mas- ter’s Thesis. Helsinki University of Technology, 1997. [138] V. Turunen, Pseudodifferential calculus on compact Lie groups and homoge- neous spaces. Helsinki University of Technology, PhD Thesis, 2001. Bibliography 691

[139] V. Turunen, Pseudodifferential calculus on the 2-sphere. Proc. Estonian Acad. Sci. Phys. Math. 53 (2004), 156–164. [140] V. Turunen and G. Vainikko, On symbol analysis of periodic pseudodifferen- tial operators. Z. Anal. Anw. 17 (1998), 9–22. [141] G. Vainikko, Periodic integral and pseudodifferential equations. Espoo. Helsinki University of Technology, 1996. [142] G. Vainikko, Personal communication, 1997. [143] G.M. Vainikko and I.K. Lifanov, Generalization and use of the theory of pseudodifferential operators in the modeling of some problems in mechanics. (Russian) Dokl. Akad. Nauk 373 (2000), 157–160. [144] G.M. Vainikko and I.K. Lifanov, The modeling of problems in aerodynamics and wave diffraction and the extension of Cauchy-type integral operators on closed and open curves. (Russian) Differ. Uravn. 36 (2000), 1184–1195, 1293; translation in Differ. Equ. 36 (2000), 1310–1322. [145] J. V¨ais¨al¨a, Topologia II. Limes, 1987. [146] N. Th. Varopoulos, Analysis on Lie groups . Journal of Functional Analysis 76 (1988), 346–410. [147] N.J. Vilenkin, Special Functions and the Theory of Group Representations. American Mathematical Society, 1968. [148] N.R. Wallach, Harmonic Analysis on Homogeneous spaces. Marcel Dekker Inc., 1973. [149] H. Whitney, Differentiable manifolds , Ann. Math. 37 (1936), 645–680. [150] H. Widom, A complete symbolic calculus for pseudodifferential operators , Bull. Sci. Math. (2) 104 (1980), no. 1, 19–63. [151] M.W. Wong, An introduction to pseudo-differential operators. Second edi- tion. World Scientific Publishing Co., Inc., 1999. [152] K. Yosida, Functional Analysis. Reprint of the sixth (1980) edition. Springer- Verlag, Berlin, 1995. [153] D.P. Zelobenko, Compact Lie Groups and Their Representations. American Mathematical Society, 1973. [154] A. Zygmund, Trigonometric series. Cambridge University Press, 1959.

Notation

Part I L(V, W ), L(V ), 82 Ker( A), Im( A), 82 ∅, x ∈ A, x 6∈ A, A ⊂ B, 10 σ(A), 82 N, N , Z, Q, R, C, Z+, R+, 10 0 x 7→ k xk , kxk, 92 A ∪ B, A ∩ B, A = B, 10 X C(K), 93 P(X), 10 B (x, r ), B(x, r ), 94 Ac, 11 X kAk , 94 f| , 13 op A L(X, Y ), L(X), 94 ∼, 14 V ′, L(V, K), 94 min, max, inf, sup, 15 LC (X, Y ), LC (X), 95 lim inf, lim sup, 16 Γ( f), 99 ∈ Xj, 18 Qj J hx, y i, 103 card( A), 20 x⊥y, M⊥N, 103 |A|, 20 P , P (x), 105 d(x, y ), 26 M M M ⊥, M ⊕ M ⊥, 106 Br(x), Bd(x, r ), 26 M1 ⊕ M2, j∈J Hj , 106 dp, d2, d∞ 27 L S , Tr( A), 111 C([ a, b ]), B([ a, b ]), 27 1 S , hA, B i , 112 d(A, B ), 28 2 S2 ∞ kAkHS , 112 (xk)k=1 , 29 d a ⊗ b, X1 ⊗ · · · ⊗ Xr, 83 lim xk = p, xk → p, xk −−−−→ p 29 k→∞ k→∞ A ⊗ B, 84 ∗ τ, τ , 31 X ⊗π Y , X⊗πY , 90 τd, τ(d), 32 X ⊗ε Y , X⊗bεY , 91

τA, 33 j∈J aj, 115b P∗ τ1 ⊗ τ2, 34 m , 116 int d(A), ext d(A), ∂d(A), 35 M(ψ), 118 Vτ (x), V(x), 36 Σ( τ), 119 NEFIS( X), 72 µ-a.e., 137

K, 79 f ∼µ g, 137 Kn, 80 f +, f −, 141 V X , 80 f dµ, 143 R p ∞ span( S), 80 L (µ), L (µ), kfkLp(µ), 152 694 Notation

+ − m n m n ν , ν , |ν|, 159 Op( A (T )), Op( Aρ,δ (T )),341 µ⊥λ, 161 Op( A∞(Tn)), Op( A−∞(Tn)), 341 1A, 191 S(Rn), 224

[A, B ], 192 Ker s, 382 σA(x), 192 sing supp, 383 A =∼ B, Hom( A, B),195 Hs(x), Hs(U), 383 Spec( A), 208 sing supp t, 383 ρ(x), 205 R(λ), 204 Operators, etc. Gel, Gel −1, 216 cl X(·), cl (·), U 7→ U, 330 H(Ω), 217 k · k X , k · k L(X,Y ), 330 u 7→ u, 302 Part II τx, Rb, 244 (·, ·)L2(Tn) = ( ·, ·)H0 (Tn), 302, 309 Spaces, sets s Tn + k · k H ( ), 307 Z , N0, Z, R, C, 298 ϕs, 308 U| W , 330 (·, ·) s Tn , 308 τ , ( X, τ ), 330 H ( ) X X h· , ·i , 308 (u ) ∈Z+ , 330 j j A∗, A(∗B), A(∗H), 309 L(X, Y ), L(X, Y ), 330 k · k , 309 supp, 239 s,t △, △ , 310 Ker, Im, 381 ξ △, △ , 310 Rn/Zn, 300 ξ k · k p , 319 Cm, C∞(Tn), 300 ℓ 1 δ , 322 S1, 299 j,k k (k) (−k) Tn = Rn/Zn, 299 ∂x, ∂x , ∂x , 327 L2, L2(Tn), 302 σ, A 7→ σA, 335 TrigPol( Tn), 305 Op, σ 7→ Op( σ), 335 Hs(Tn), 307 a 7→ a, 342, 346 Hs,t (Tn × Tn), 309 a 7→ ba1, a 7→ a2, 346 ∞ C (Tn × Tn), 309 f 7→ bf, FRn ,b 222 − ∞ n n 1 C (T × Z ), 338 FRn ,b 225 −1 m n n m n n FTn , F n , 301 S (T × Z ), Sρ,δ (T × Z ), T ∞ n n −∞ n exp, 368 Sρ,δ (T × Z ), S (T ), 338 ∗ ∗ ∗ Sm(Rn × Rn), 260 a , a( B), a( H), 370 m n n Sρ,δ (R × R ), 261 [·, ·], 371 m n n m n n Op( S (T × Z )), Op( Sρ,δ (T × Z )), Lj, Rk, 423 338 [·, ·]θ, 385 ∞ n −∞ n Op( S (T )), Op( S (T )), 338 σA, 262 m n m n α A (R ), Aρ,δ (R ), 275 Dx , 327 m n m n −∞ n (α) A (T ), Aρ,δ (T ), A (T ), 341 (−Dy) , 394 Notation 695

Other notation HOM( G1, G 2), 448 1 1 p.v. , , 238 ∈ φ|Hj , 450 x x±i0 Lj J A|W , 330 Haar( f), 459

A(x)|x=x0 , 330 PG/H , 462 τ uj −→ u, 330 Haar G/H , 463 Ind, 381 G, 468 dim, codim, 381 TrigPol(b G), 474 z, 302 L2(G), 477 G hξi, 221, 300 Res Hbψ, 482 (j) ξ , 313 Cφ(G, H), 484 (j) S , kj , 321 Ind GH, 484 k  φ K, 340, 350 exp( X), 492 m,ρ,δ ∼, ∼m, ∼ , 342 Lie K(A), 499 [·], 342 Lie (G), g, 499 ∞ gl (n, K), o(n), so (n), u(n), su (n), 500 σ ∼ j=0 σj , 352 P ∞ SL( n, K), sl (n, K), 501 Op (σ) ∼ j=0 Op (σj ), 352 P Ad( A)X, 505 pj → ∞, 386 α!, 223 ad( X)Y , 506, 508 α ≤ β, 223 U(g), 507 |α|, 223 Part IV Dα, 224 L, 225 Re G, 530 Hξ, ξ : G → U (Hξ), 530 Part III f(ξ), 531 K, 430 φbu, 531 Aut( V ), 431 DY f, 532 Aff( V ), 431 f(ξ)mn , 533 xA , Ax , AB , A0, A−1, An, A−n, 432 δbmn , 533 H < G , H ⊳ G, 432 LG, L, 534 ′ Z(G), 432 D (G), 534 GL (n, R), O(n), SO (n), 433 h· , ·i G, 535 s GL (n, C), U(n), SU (n), 433 H (G), 535 ξ G/H , 433 H , 536 H\G, 434 dim( ξ), 536 λξ, 537 Gq, 437 U(H), 438 M(G), 537 2 b πL, πR, 439 L (G), 537 φ ∼ ψ, 441 (·, ·)bL2(G), 537 b τG/H , 447 hξi, λ[ξ], 538 696 Notation

S(G), pk, 539 S′(bG) 543 h· , ·i bG, 543 2 − 1 p b p p( p 2 ) L (G), ℓ G, dim , 546 p b b Lk(G), 550 KA(bx, y ), LA(x, y ), RA(x, y ), 550 l(f), r(f), 551, 579 ∂α, 534, 560 σA(x, ξ ), 552 fφ, Aφ, 556 uL, uR, 556 hA, B iHS , || A|| HS , || A|| op , 559 α △ξ , 564 △q, 564 m Ak (M), 566 m m Σ (G), Σ k (G), 575 πL, πR, 580 RA(x), LA(x), 582 la, la(x), rA, rA(x), 583 D(G), 591 SU(2), 599 H, 603 Sp( n), 606 Sp( n, C), 606 w1, w 2, w 3, 607 Y1, Y2, Y3, 607 D1, D 2, D 3, 609 ∂+, ∂ −, ∂ 0, 611 Vl, Tl, 612 l l l t , tmn , Pmn , 617 t−− , t−+, t+−, t++ , 621 f(l)mn , 632 σbA(x, l ), σA(x, l )mn , 632

σ∂+ , σ∂− , σ∂0 , 634 Sm(SU(2)), 656 Sm(S3), 661 ′ DL1 (M), 668 pE→B, 668 K\G, 669 Index

∗-algebra, 213 involution, 213 Ψ( M), 423 isomorphism, 195 Diff( M), 419 Lie, 498 of pseudo-differential operators quotient, 194, 199 n on T , 380 radical of, 193, 211 ∗-homomorphism, 213 semisimple, 193 hξi on a group, 538 tensor product, 196 tl on SU(2), 633 topological, 196 µ-almost everywhere, 137 unit, inverse, 191 µ-integrable, 143 unital, 191 ∂α on groups, 560 universal enveloping, 507 σ-algebra, 119 algebra of periodic ΨDOs, 367, 380 Abel–Dini theorem, 386 algebra reformulation, 518 action associativity diagram of, 518 free, 668 co-algebra, 519 left, right, 437 co-associativity diagram, 519 linear, 437 multiplication mapping, 518 of a group, 436 tensor product, 518 transitive, 437, 669 unit mapping, 518 adjoint operator, 107 algebraic Banach, 101 basis, 80 on a group, 569, 591 dimension, 81 adjoints (Banach, Hilbert), 309 number, 24 Ado–Iwasawa theorem, 508 tensor product, 84 affine group, 431 algebra, 191 almost orthogonality lemma, 95 ∗-algebra, involutive, 213 amplitude Spec( A), spectrum of, 208 of adjoints, 370 Banach, 200 of periodic integral operator, 388 C∗-algebra, 213 operator, 275, 341 character of, 208 toroidal, 340 m n m n commutative, 191 amplitudes A (R ), Aρ,δ (R ), 275 m n m n derivations of, 499 amplitudes A (T ), Aρ,δ (T ), 340 homomorphism, 195 Arzel`a–Ascoli theorem, 57 Hopf, 520 asymptotic equivalence, 342 698 Index asymptotic expansion, 352, 353 bounded inverse theorem, 99 of adjoint, 279, 370 C∗-algebra, 213 of parametrix, on a group, 577 Calder´on–Zygmund covering lemma, of parametrix, toroidal, 380 257 of product, 371 canonical mapping of a Lie algebra, of transpose, 280, 369 507 asymptotic sums, 351 Carath´eodory condition, 125 atlas, 416 Carath´eodory–Hahn extension, 123 automorphism, 430 cardinality, 20 inner, 531 Cartan’s maximal torus theorem, space Aut( V ), 431 481 automorphism group, 431 Cartesian product, 12, 18, 71 Axiom of Choice, 18, 25, 73 Casimir element, 510 for Cartesian products, 18 Cauchy’s inequality, 229 Baire’s theorem, 96 Cauchy–Schwarz inequality, 103, balls Br(x), Bd(x, r ), 26 229, 230 Banach chain, 15 adjoint, 101 character algebra, 200 of a representation, 479 duality, 308 characterisation of S−∞(Tn), 348 fixed point theorem, 43 characteristic function, 13, 135 injective tensor product, 91 characters, 582 projective tensor product, 91 Chebyshev’s inequality, 148 , 94 choice function, 17 dual of, 101 theorem, 99 reflexive, 102 closure, 36 Banach–Alaoglu theorem, 99 closure operator, interior operator, in Hilbert spaces, 109 37 in topological vector spaces, 89 co-algebra, 519 Banach–Steinhaus theorem, 97 monoid, 521 barrel, 90 co-induced basis family, 14 algebraic, 80 topology, 69 orthonormal, 110 commutant, 198 Bernstein’s theorem, 306 commutator, 192, 371 bijection, 13 commutator characterisation bilinear mapping, 83 Euclidean, 414 Borel on a group, 566 σ-algebra, 119 on closed manifolds, 421 measurable function, 135 toroidal, 424 sets, 119 , 101 Borel–Cantelli lemma, 122 complete topological , boundary, 36 86 Index 699 completion, 44 de Morgan’s rules, 12 of a , 86 density, 36 component, 448 derivations of operator-valued composition, 13 symbols, 587 composition formula derivatives and differences, 325 Euclidean, 271 diameter, 28 for Fourier series operators, 394, difference operators 397 forward, backward, 310 on a group, 567, 568 on SU(2), △q, △+, △−, △0, 636 toroidal, 371 on SU(2), formulae for, 638 continuity on a group, 564 metric, 30 Dirac topological, 46 delta, 239, 243 uniform on a group, 452 delta comb, 306, 364 continuum hypothesis, 26 direct sum, 101, 106 generalised, 26 algebraic, 440 contraction, 43 of representations, 450 convergence discrete almost everywhere, 138 cone, 389 almost uniform, 138 p fundamental theorem of calculus, in L (µ), 155 314 in measure, 138 integration, 314 in metric spaces, 29 partial derivatives D(α), 327 in topological spaces, 32 x polynomials, 313 metric uniform, 42 of a net, 77 Taylor expansion, 315 pointwise, 41, 138 disjoint family, 119 uniform, 42, 138 distance, 26 , 89 between sets, 28 convolution, 228 distribution function, 255 left, right, l(f), r(f), 579 distributions D′(Ω), 242 associativity of, 228 ′ non-associativity of, 245 E (Ω), 242 ′ n of distributions, 244 D (T ), periodic, 304 of linear operators, 520 on manifolds, 419 of sampling measures, 456 periodic, 307, 308 ′ on a group, 478, 532 summable, DL1 (M), 668 translations of, 246 dual convolution kernel, left, right, 582 algebraic, 84 convolution operators, 551, 579 Banach, Hilbert, 308 Cotlar’s lemma, 406 of Lp(µ), 170 cover, 49 of a Banach space, 101 locally trivialising, 668 second, 102 cyclic vector, representation, 450 space, 85 700 Index

unitary, 468 Fourier transform duality f(l)mn , on SU(2), 632 h· , ·i G, 535 andb rotations, 227 h· , ·i G, 543 Euclidean, 222 b inverse, on S′(G), 545 Egorov’s theorem, 139 p b ellipticity, 376 inverse, on L (G), 548 on a group, 577 matrix, 533 b embedding, 309 multiplication formula, 226 embedding theorem, 294 of Gaussians, 226 endomorphism, 430 of tempered distributions, 233 ′ equicontinuous family, 57, 85 on D (G), 545 equivalence relation, 14 on L1(G), 548 Euler’s angles on Lp(G), 548 ′ on S3, 604 on D (Tn), 305 on SO(3), 597 on a group, 475 on SU(2), 601 on group G, 531 Euler’s identity, 284 toroidal, periodic, 301 exponential coordinates, 500 Fr´echet space, 87 exponential of a matrix, 492 Fredholm extreme set, 89 integral equations, 44 operator, 381 family, 9 freezing principle, 288 family induced, co-induced, 14, 134 Frobenius reciprocity theorem, 488 Fatou’s lemma, 146 reverse, 147 Fubini theorem, 187 Fatou–Lebesgue theorem, 151 Fubini–Tonelli theorem, 186 fiber, 668 function, 12 fiber bundle, 668 M-measurable, Borel measurable, principal, 668 Lebesgue measurable, 135 finite intersection property, 50, 72 H¨older continuous, 306 Fourier coefficients harmonic, 293 on a group, 475 holomorphic, 217 Fourier coefficients, series, 302 negative part of, 141 Fourier inversion periodic, 300 global, 580 positive part of, 141 Fourier inversion formula simple, 141 Euclidean, 225 test, 88 on S′(Rn), 238 weakly holomorphic, 204 on S(Zn), C∞(Tn), 301 functional Fourier series Haar, 454–460 on L2(Tn), 302 linear, 82 on a group, 475 positive, 175, 453 Fourier series operator, 393, 407 positive, in C ∗-algebra, 216 Index 701 functional calculus at the normal symmetric, 431 element, 216 topological, 445 transformation, right, left, 668 Gelfand unitary U( n), 433, 438 theorem, 1939, 203 theorem, 1940, 208, 210 H¨older’s inequality, 153, 230 theory, 207 converse of, 173 topology, 209, 517 discrete, 319 transform, 209, 517 for Schatten classes, 113 Gelfand–Beurling spectral radius general, 231 formula, 205 Haar Gelfand–Mazur theorem, 205 functional, 454–460 Gelfand–Naimark theorem, 214 integral, 454 commutative, 215 measure, 454 graph, 99 Haar integral group, 430 on SO(3), 599 SU(2), 599 on SU(2), 605 SO(2), 596 Hadamard’s principal value, 238 SO(3), 596 Hahn decomposition, 161 Sp( n, C), 606 Hahn–Banach theorem, 96 Sp( n), 606 in locally convex spaces, 88 U(1), 595 Hamel basis, 80 unitary, U(H), 438 Hausdorff action, 436 maximal principle, 18, 73 affine, 431 centre of, 432 space, 53 commutative, Abelian, 431, 491 total boundedness theorem, 86 compact, 451 Hausdorff–Young inequality, 236, finite, 431 304 general linear GL( n, R), on G and G, 548 GL( n, C), 433 Heaviside function,b 239 homomorphism, 435 Heine–Borel property, 90 infinitesimal, 498 Heine–Borel theorem, 59 isomorphism, 435 Hilbert Lie, 491 duality, 308 linear Lie group, 491 space, 103 locally compact, 451 Hilbert–Schmidt, 559, 662 orthogonal O( n), 433 operators, 112 permutation, 432 spectral theorem, 109 product, 431 homeomorphism, 48 representation of, 439 homomorphism, 195, 430 special linear SL (n, K), 501 continuous, 448 special orthogonal SO( n), 433 differential, 502 special unitary SU( n), 433 space HOM( G1, G 2), 448 702 Index

Hopf algebra, 520 isomorphism, 195, 430 “everyone with the antipode” canonical, 308 diagram, 521 intertwining, 531 and C∗-algebra, 524 isometric, 48 antipode, 520 isotropy subgroup Gq, 437 co-multiplication and unit Jacobi identity, 361, 498 diagram, 521 for Gaussians, 361 for compact group, 523 Jordan decomposition, 159, 160 for finite group, 522 multiplication and kernel co-multiplication diagram, 521 of a linear operator, 82 multiplication and co-unit kernel, null space, 430, 435 diagram, 521 kernels la, la(x), rA, rA(x), 583 tensor product, 520 Killing form, 509 Hopf fibration, 673 Krein–Milman theorem, 89 hyperbolic equations, 410 Krull’s theorem, 193 hypoellipticity, 384, 385 Kuratowski’s closure axioms, 37 ideal Laplace operator, 225 spanned by a set, 193 on SU(2), 611, 625 two-sided, maximal, proper, 193 on a group, 512, 534 on a group, symbol of, 554 index, 388 law of trichotomy, 21 of Fredholm operator, 381 Lebesgue index sets, 11 Lp(µ)-norm, 152 induced Lp(µ)-spaces, 154 family, 14 –B.Levi monotone convergence induced representation space theorem, 144 Ind GH, 484 φ conjugate, 153 injection, 13 covering lemma, 61 inner product, 103 decomposition of measures, 168 on V ⊗ W , 84 differentiation theorem, 252, 253 integral, 143 dominated convergence theorem, Haar, 454 150, 222 Lebesgue, 143 integral, 143 Pettis, weak, 90, 482 measurable function, 135 Riemann, 151 measure, 128 integration measure, translation and rotation discrete, 314 invariance, 129 interior, 36 measurelet, 117 interpolation theorems, 385 non-measurable sets, 133 invariant outer measure, 117 p vector fields, 500 space L (µG) on a group, 460 isometry, 443 left quotient H\G, 434 Index 703

Leibniz formula Marcinkiewicz’ interpolation asymptotic, 571 theorem, 256 discrete, 311 maximum, minimum, supremum, Euclidean, 249 infimum, 15 on an algebra, 499 measure, 121 LF-space, 88 absolutely continuous, 163 Lie action-invariant on G/H , 463 algebra, 498 Carath´eodory–Hahn extension of, algebra sl (n, K), 501 123 algebra homomorphism, 498 Haar, 454 algebra of a Lie group, 499 Hahn decomposition of, 161 algebra, canonical mapping, 507 Jordan decomposition of, 159 algebra, semisimple, 509 Lebesgue, 128 algebras gl (n, K), o(n), so (n), Lebesgue decomposition of, 168 u(n), su (n), 500 outer, 116 group, 491 probability, 122 group, dimension of, 499 product of, 181 group, exponential coordinates, Radon–Nikodym derivative of, 500 162 group, linear, 491 sampling, 455 group, semisimple, 509 semifinite, 158 subalgebra, 498 signed, 158 lifting of operators, 673 variations of, positive, negative, linear operator total, 159 bounded, 94 measure space, 122 compact, 95 σ-finite, 164 norm of, 94 Borel, 122 Liouville’s theorem complete, 122 for harmonic functions, 293 finite, 122 for holomorphic functions, 217 measurelet, 116 Littlewood’s principles, 142 measures locality, 265, 382 mutually singular, 161 logarithm of a matrix, 495 metric, 26 Luzin’s theorem, 141 discrete, 27 Euclidean, 27 manifold, 65, 417 interior, closure, boundary, 35 closed, 419 subspace, 27 differentiable, 66 sup-metric, d∞, 27 orientable, 418 metric space paracompact, 423 complete, 40 mapping, 12 sequentially compact, 58 continuous, uniformly continuous, totally bounded, 61 Lipschitz continuous, 49 metrics measurable, 134 equivalent, 49 704 Index

Lipschitz equivalent, 33 operators Minkowski’s functional, 87 Hilbert–Schmidt, 112 Minkowski’s inequality, 153, 231 , 111 for integrals, 188 operators ∂+, ∂ −, ∂ 0, 611 mollifier, 251 applied to tl, 628 monoid, 456 symbols of, 634 , 90 in Euler angles, 611 multi-indices, 223 order multiplication of distributions, 238 partial, 15 total, linear, 15 Napier’s constant, 40 orthogonal projection, 105 neighbourhood, 29, 36 outer measure, 116 net, 77 Borel regular, 124 Cauchy, 86 metric, 125 neutral element, inverse, 431 product of, 181 norm, 92 equivalent, 93 parallelogram law, 105 operator, 94 parametrix, 287, 378, 380 trace, 111 on a group, 577 normal Parseval’s identity divisors, 432 on Rn, 236 element, 216, 517 on a group, 475, 476, 538 subgroup, 432 partition of unity, 56 , 92 path, 67 numbers, 10 Pauli matrices, 607 algebraic, 24 Peetre’s Stirling, 321 inequalities, 321 theorem, 266 open mapping, 98 periodic open mapping theorem, 98 Schwartz kernel, 336 operator Taylor expansion, 328 α D , 224 periodic integral operator, 387 α (α) Dx , Dx , 327 periodicity, 300 (α) ( − Dy) , 394 periodisation, 360 adjoint, 107 compactly supported classical, 355 perturbations, 366 compact, 193 of operators, 363 intertwining, 441 of symbols, 365 left-invariant, right-invariant, 551 Peter–Weyl theorem, 470 linear, 82 for Tn, 471 order of, 414, 420 left, 471 properly supported, 280 Pettis integral, 90, 482 self-adjoint, 107 Plancherel’s identity, 302 operator norm, 94 in , 110 Index 705

on Rn, 236 quantum numbers, 630, 661 on a group, 475, 476 quaternions, 603, 660 point quotient accumulation, 36, 52 algebra, 194, 199 fixed, 43 left H\G, 434 isolated, 36 right G/H , 433 Poisson summation formula, 361 topology, 69, 199 polynomial topology on G/H , 447 discrete, 313 vector space, 82 trigonometric, TrigPol( G), 474 trigonometric, on Tn, 305 Radon–Nikodym derivative, Pontryagin duality, 469 theorem, 162, 164 power set P(X), 10 relation, 12 preimage, 30 Rellich’s theorem, 295 preimage, image, 13 representation principal symbol, 352 tl on SU(2), 617 principle regular, left, right, 551, 580 convergence, 234 adjoint, of a Lie algebra, 506 Littlewood, 142 adjoint, of a Lie group, 505 uniqueness, 234 cyclic, 450 product decomposition of, 468 of measures, outer measures, 181 dimension dim ( ξ), 530 topology, 34, 40, 72 dimension of, 439 product group, 431 direct sum of, 450 projection PG/H , 462 equivalent, 441 pseudo-differential operator induced, 484 local, 414 irreducible, 440 on a manifold, 418 matrix, 469 periodic, continuity of, 343 multiplicity of, 488 toroidal, 338 of a group, 439 pseudo-differential operators regular, left, right, πL, πR, 439, m Ψ (G), 573 470 m Ψ (M), 418 restricted, 440 Ψm(SU(2)), Ψ m(SU(2)), 528 G space Ind φ H, 484 on a group, 553 space Rep( G), 530 pseudolocality, 265, 383 strongly continuous, 449 pushforwards, φ–, 556 topologically irreducible, 450 Pythagoras’ theorem, 103 unitary, 439 quantization unitary matrix, 439 on Rn, 276 resolvent mapping, 204 on a group, 552 restriction, 13 operator-valued, 583 Riemann integral, sums, 151 toroidal, 336 Riemann–Lebesgue lemma, 223 706 Index

Riesz linearly independent, 80 almost orthogonality lemma, 95 open, 29 representation theorem, 107 open, closed, 31 topological representation well-ordered, 16 theorem, 175, 177 singular support, 243, 245, 383 Riesz–Thorin interpolation theorem, small sets property, 85 158 small subgroups, 496 right quotient G/H , 433 smooth mapping, 417 right transformation group, 668 smoothing operators, 266 Russell’s paradox, 11 smoothing periodic ΨDOs, 347 Sobolev spaces, 293 scaling, 241 Hs(G), 535, 542 Schatten class, 113 Lp(G), 550 Schr¨oder–Bernstein theorem, 21 k Lp(Ω), 247 Schr¨odinger equation, 412 k b Lp(Tn), 375 Schur’s lemma, 284, 443 s biperiodic, 309 Schwartz kernel, 92, 340, 350, 550, localisation, Lp(Ω) , 248 592 k loc on manifolds, Hs(M), 419 periodic, 336 toroidal, Hs(Tn), 307 Schwartz kernel theorem, 92 space , 87, 224 C∞(Tn × Zn), 338 S(Rn), 224 Ck(M), C∞(M), 417 S(Zn), 300 C∞(Ω), 239 Schwartz’ impossibility result, 238 0 L2(Tn), 302 semigroup, 456 2 , 92 L (G), 477 p Rn separability, topological, 36 Lloc b( ), 241 p separating points, 55, 75 L (G), 546 p n , 29 L (Rb ), 221 p n Cauchy, 40 L (R ), interpolation, 232 p n generalised, 77 L (T ), 304 sequential density of functions, 240 P ol 1(SU(2)), 657 set K-vector, 79 m directed, 77 Ψ (M), 418 D′(G), 534, 591 sets, 9, 10 ′ ψ-measurable, 118 D (M), 419 balanced, 80 D(G), 591 Borel, 119 D(M), 419 ′ bounded, 28 S (G), 543 compact, 49 M(bG), 537 convex, 80 S(Gb), 539 elementary, 116 Diff(b M), 419 extreme, 89 barreled, 90 Lebesgue non-measurable, 133 base, 668 Index 707

homogeneous, 669 operator-valued, left, right, 583 measure, 122 principal, 352 metric, 26 toroidal, 335, 336 quotient, 14 symbol class simply-connected, 504 m n n Sρ,δ (R × R ), 261 topological, 31 m n n Sρ,δ (T × R ), 338 total, 668 m n n Sρ,δ (T × Z ), 338 span, 80 Sm(S3), 661 spectral radius formula, 205 Sm(SU(2)), 656 spectrum Σm(G), 575 of an algebra element, σ(x), 192 Σm(SU(2)), 633 of an algebra, Spec( A), 517 Σm(SU(2)), Σ m(SU(2)), 633 of an operator, σ(A), 82 0 k Stein–Weiss interpolation, 549 Taylor expansion Stirling numbers, 321 biperiodic, 328 recursion formulae, 322 discrete, 315 Stone–Weierstrass theorem, 63 on a group, 561 subalgebra periodic, 327, 328 involutive, 63 tempered distributions subcover, 49 S′(Rn), 233 subgroup, 432 S′(Zn), 300 trivial, 432 tensor product subnet, 77 algebra, 196, 518 subspace algebraic, 84 compact, 50 Banach, injective, 91 invariant, 440 Banach, projective, 91 metric, 27, 34 Hopf algebra, 520 trivial, 80, 440 injective, 91 vector, 80 of operators, 84 subspaces orthogonal, 103 of spaces, 83 sum, infinite sum, 115 projective, 90 summation by parts, 313 spaces, dual of, 84 summation on SU(2), 630 test functions, 88, 300 support, 55, 243 Tietze’s extension theorem, 71 surjection, 13 Tihonov’s theorem, 73 Sweedler’s example, 526 topological symbol algebra, 196 classical, 283, 355 equivalence, 48 elliptic, 289 group, 445 Euclidean, σA, 262 interior, closure, boundary, 36 homogeneous, 283 property, 48 of periodic ΨDO, 357 vector space, 85 on a group, 552 zero divisor, 203 708 Index topological approximation trace, trace class, trace norm, 111 of Lebesgue measurable sets, 131 transfinite induction, 17 of measurable sets, 126 mathematical induction, 17 topological space transpose, 279 compact, 49 transposed operator complete, 86 on a group, 570, 591 completion of, 86 triangle inequality, 26, 104 connected, disconnected, 66 trigonometric polynomials Hausdorff, 53 TrigPol( G), 474 locally compact, 49 Tn locally convex, 87 TrigPol( ), 305 paracompact, 423 uncertainty principle, 241 path-connected, 67 uniform boundedness principle, 97 separable, 36 unit, 191 totally bounded, 86 unital algebra, 191 topology, 31 F-induced, 71 unitary dual G, 468, 530 base of, 39 universal envelopingb algebra, 507 co-induced, 69 as Hopf algebra, 526 discrete, 75 universality induced, 48 of enveloping algebra, 507 injective tensor, 91 of permutation groups, 436 metric, 36, 39 of unitary groups, 491 metric, canonical, 32 Urysohn’s lemma, 55 metric, comparison, 32 smooth, 254 metrisable, 74 norm, 94 vector space, 79 on R2, 34 Banach, 94 product, 34, 40, 71 dimension of, 81 projective tensor product, Fr´echet, 87 π-topology, 90 Hilbert, 103 quotient, 69, 199 inner-product, 103 quotient on G/H , 447 LF-, 88 relative, 33 locally convex, 87 second countable, 39 Montel, 90 strong operator, 449 normed, 92 subbase, subbasis, 39 nuclear, 92 weak, 88, 109 weak ∗, 88, 99 quotient, 82 toroidal topological, 85 amplitude, 340 vectors torus, 299 linearly independent, 80 inflated, 362 orthogonal, orthonormal, 103 tower, 19 Vitali’s convergence theorem, 156 Index 709 wave front set, 390 weak derivative, 246 , 88, 109 weak type ( p, p ), 256 weak ∗-topology, 88, 99 Weierstrass theorem, 62 Well-ordering principle, 17, 25 Whitney’s embedding theorem, 418 Young’s inequality, 187 discrete, 320 for convolutions, 231 general, 232 on Rn, 229 Zermelo–Fraenkel axioms, 26 Zorn’s lemma, 19