The Harald Bohr Centenary Proceedings of a Symposium held in Copenhagen April 24-25, 1987 Edited by C . BERG and B . FUGLED E Matematisk-fysiske Meddelelser 42 : 3 Det Kongelige Danske Videnskabernes Selskab The Royal Danish Academy of Sciences and Letters Commissioner : Munksgaard Copenhagen 1989 QC Det Kongelige Danske Videnskabernes Selskab 1989 Printed in Denmark by Bianco Lunos Bogtrykkeri A/ S ISSN 0023-3323 . ISBN 87-7304-193-9 Preface These are the proceedings of the symposium held in Copenhagen, April 24-25, 1987, o n the occasion of the centenary of Harald Bohr (April 22, 1887 -January 22, 1951) . The symposium was arranged by the Danish Mathematical Society which appointed a n organizing committee consisting of C . Berg, B . Fuglede, and L .-E. Lundberg. The scientific programme comprised 13 lectures of which 11 were given by invited foreign scholars . Former colleagues, students and friends of Harald Bohr profited from thi s occasion to commemorate the work and personality of a Danish mathematician of hig h international rank . The organizing committee was happy to have the symposium opened by the rector of the University of Copenhagen, O. Nathan, a former student of Harald Bohr during those years of the war which they had to spend in Sweden . The main achievement in the mathematical works of Harald Bohr - the theory o f almost periodic functions - has developed in several directions during the past sixt y years, and the papers in the present Proceedings will show some of these trends . We take the opportunity of thanking all the participants for their presence and thei r scientific contribution to the symposium . The Danish Mathematical Society gratefully acknowledges financial support fro m The Augustinus Foundation The Carlsberg Foundatio n The Danish lblinistery of Educatio n The Danish Natural Science Research Counci l The Otto Mønsted Foundation . The Organizing Committee List of Participants Faraut, J. (Strasbourg ) Address Copenhagen unless otherwise indicated photo no . 2 7 Fegan, H. D . (Albuquerque ) photo no . 22 Amerio, L . (Milano) Bohr, L . photo no . 28 Felsager, B . photo no . 45 Bohr, O. Fenchel, W Alsholm, P. (Lyngby) Bohr, 'T. Fink, A . M . (Ames ) Andersen ; C . Booss, B . B . (Roskilde) photo no. 40 Andersen, E . S . photo no . 16 Branner, B . (Lyngby) Flensberg, K . Andersen, I . photo no . 5 photo no . 61 Flensted Jensen, M . Andersen, K . S. Brinch, P. photo no . 32 Andersen, S . A . Brøndsted, A. Foxby, H .-B . Andersen, T. F. Buch, K . photo no . 8 Fuglede, B . photo no . 23 Ballhausen, C . J. Christensen, E. photo no . 25 Gliese, H . Bang, T . photo no. 48 Christensen, J. P. R. Grubb, G . photo no . 38 Bauer, H . (Erlangen) photo no. 51 Haagerup, U . (Odense ) photo no . 15 Christensen, L . Handest, F. photo no. 19 Bellissard, J. (Marseille) Christiansen, B . Hansen, F . photo no . 3 7 photo no. 47 Christoffersen, T. (Sorø) Hansen, V. L . (Lyngby ) Berg, C . photo no . 43 De Vries, J. (Amsterdam) photo no . 26 Berglund, J. E (Richmond) photo no . 6 Harremoës, P. photo no . 21 photo no. 36 Durhuus, B . Hedberg, L . I . (Linköping ) Bertelsen, K . Dylander, J. photo no . 63 Bohr, Aa. Eymard, P. (Nancy) Heefelt, M . B . (Roskilde ) Bohr, H . photo no . 60 photo no . 4 Bohr, J. Falgaard, P. V. (Svendborg) Heiede, T. Photo by P. AIne s Helgason, S . (Boston) Mantini, L . (Oklahoma) Ressel, P. (Eichstätt ) photo no. 53 photo no . 52 photo no . 24 Hewitt, E . (Seattle) Markvorsen, S . (Lyngby) Rugh, H . H . photo no . 56 photo no. 39 photo no . 29 Rugh, S . E . photo no . 31 Hörmander, L . (Lund) Mejlbro, L . (Lyngby) Sasvari, Z. (Dresden ) photo no . 2 Meyer, H. photo no . 18 photo no . 11 Jakobsen, H . P. photo no . 11 Milnes, P. (Ontario) Schlichtkrull, H . photo no . 33 Jensen, A . photo no . 55 Schmidt, A . L . photo no . 42 Jensen, B . S . Nathan, O . Schmidt, C . Jensen, J . S . Nielsen, H . B . photo no . 49 Schmidt, S . Jensen, U . K . Nikoltjeva-Hedberg; M . Solovej, M . Jessen, B . photo no . 34 (Linköping) photo no . 62 Stetkær, H . (Århus ) Johansen, T. B . Olesen, D. photo no . 41 Svensmark, H . Jørsboe, O . (Lyngby) Pedersen, E D . (Lyngby) Terp, M . photo no . 3 photo no . 59 Thill, M . (Luxembourg ) Kahane, J.-P. (Orsay) Pedersen, G . K . Tolnø, J . photo no . 46 Pedersen, H . Topsøe, F. photo no . 1 0 Kehlet, E . Pedersen, N . V. photo no . 58 Tornehave, H . photo no . 50 Kjeldsen, T. H . photo no . 54 Pedersen, S . Torpe, A . M . (Odense ) Kokholm, N . J. photo no . 7 Praetorius, H . W. Vedelseby, M . Larsen, M . E . photo no . 35 Prouse, G . (Milano) Wachmann, C . Lauritzen, N. photo no . 44 Waidtløw, Ma . (Lyngby ) Lindhardt, J. (Århus) Rand, D . (Warwick) Waidtløw, Mo . (Lyngby ) Lundberg, L .-E . photo no . 57 photo no . 20 Winstrøm-Olsen, B . Lützen, J. photo no. 30 Reimann, M . (Bern) Wright, D . (Oklahoma ) Lønsted, K . photo no . 12 photo no. 1 7 Madsen, T. G. Contents Tornehave, H.: Harald Bohr. Professor and head of department 9 Berg, C : Introduction to the almost periodic functions of Bohr 1 5 Amerio, L .: Almost-periodic functions in Banach spaces 25 Bellissard, j. : Almost periodicity in solid state physics and C*algebras 35 Berglund, J. F. : Dissipative weakly almost periodic functions 77 Eymard, P. : On maps which preserve almost periodicity 81 Fink, A. M .: Separated solutions of almost periodic differential equations 87 Helgason, S .: Value-distribution theory for holomorphic almost periodic functions 93 Kahane, J .-P. : Around Bohr's thesis 103 Milnes, P.: Bohr almost periodicity and functions of dynamical type 113 Prouse, G .: Almost periodic solutions of Navier-Stokes equations an d inequalities 119 Tornehave, H.: On the zeros of entire almost periodic functions 125 (This paper was not presented during the Symposium ) Lectures not published in these Proceeding s Bohr, T. : Mode-locking, circle maps and the Devil's staircase . (See Bak, P., Bohr, T., Jensen, M . H.: Circle maps, mode lockin g and chaos . In: Directions in Chaos, Vol . 2 . Edited by Hao Bai - Lin . World Scientific, Singapore 1988 ) Hewitt, E. : Marcel Riesz's theorem on conjugate Fourier series and it s descendants. (See Asmar, N., Hewitt, E . : Same title . In: Proceedings of th e Analysis Conference, Singapore 1986 . Edited by Choy, Jesudason, Lee. Elsevier Science Publishers B . V. (North- Holland) 1988) Rand, D. : Universal aperiodic attractors in dynamical systems . (Sec Rand, D . A. : Fractal bifurcation sets, renormalizatio n strange sets and their universal invariants . Proc. R. Soc. London A 413, 45-61 (1987)) Mf1VI 42 :3 9 Harald Bohr Professor and Head of Departmen t By HANS TORNEHAVE This is an attempt to tell about Harald Bohr as we knew him, "we" meaning students and junior teachers . I was his student 1935-1940 and a junior teacher at the institute til l after his death in 1951 . The nineteenhundred and thirties were the peak of Harald Bohr's career . The almost periodic functions had made him known all over the mathematical world . The original papers appeared in 1924-26 and the monograph in Ergebnisse in 1932 . He had been a professor at the Technical University since 1914, but was called to our university in 1930 . He got his new mathematical institute in 1934 at the 450' h anniversary of th e university. It was a grant from the Carlsberg Foundation, and it was built as a new win g of his brother's Institute for Theoretical Physics, which is now called the Niels Boh r Institute . Before he started in his new position at the university, Harald Bohr went on a tour t o U.S .A ., where he visited Stanford University and the Institute for Advanced Study . He was a member of the Royal Danish Academy of Sciences and Letters from 1918, and h e was the chairman of our Mathematical Society from 1935 . According to Harald Bohr the textbook in mathematical analysis known as "Bohr and Mollerup" was inspired by Jordan's Cours d'Analyse, but also Hardy : A Course of Pure Mathematics has had some influence . Johannes Mollerup has probably been somewha t underestimated and most of the details in the text-book have certainly been the result o f discussions between the authors . However, Bohr wrote very much like he talked, and we who knew him hear the echo of his voice when we read his book . Hence, we think that Bohr is responsible for the formulation, but we also know that he was extremel y willing to accept good ideas proposed by others. Bohr did not give elementary lectures at the university . The students attended th e lectures for engineering students on mathematical analysis, theoretical mechanics an d physics, while they shared the chemistry lectures with the students of medicine . The students had separate elementary lectures only in geometry and astronomy . In 1933 Bohr tried just once to lecture over his textbook for the mathematics students . The lecture went on until the fall term in 1935 . Bohr was somewhat more sedate than i n his younger days . He had given up the habit of keeping the sponge on the floor, kicking it ceilingwards and catching it neatly, when he had to erase something. He had been a top soccer player, expert dribbler and very popular. He continued playing even as a professor at a time when tailcoat and tophatwere standard equipment at the university . In the thirties it was a lot more informal, although most students wore a suit and a tie at the lectures . 10 MfM 42 :3 Bohr talked very fast and with a flat " a" reminiscent ofCopenhagen dialect. He used the blackboard in a systematic way starting upper left and finishing lower right, writin g in long horizontal lines .