Raymond E. A. C. Paley
|Died||7 April 1933 (aged 26)|
|Alma mater||University of Cambridge|
|Awards||Smith's Prize (1930)|
Paley was born in Bournemouth, England, the son of an artillery officer who died of tuberculosis before Paley was born. He was educated at Eton College as a King's Scholar and at Trinity College, Cambridge. He became a wrangler in 1928, and with J. A. Todd, he was one of two winners of the 1930 Smith's Prize examination.
He was elected a Research Fellow of Trinity College in 1930, edging out Todd for the position, and continued at Cambridge as a postgraduate student, advised by John Edensor Littlewood. After the 1931 return of G. H. Hardy to Cambridge he participated in weekly joint seminars with the other students of Hardy and Littlewood. He traveled to the US in 1932 to work with Norbert Wiener at the Massachusetts Institute of Technology and with George Pólya at Princeton University, and as part of the same trip also planned to work with Lipót Fejér at a seminar in Chicago organized as part of the Century of Progress exposition.
Paley, born in 1907, was one of the greatest stars in pure mathematics in Britain, whose young genius frightened even Hardy. Had he lived, he might well have turned into another Littlewood: his 26 papers, written mostly in collaboration with Littlewood, Zygmund, Wiener and Ursell, opened new areas in analysis.— Béla Bollobás, Littlewood's Miscellany, Foreword
Paley's contributions include the following.
- His mathematical research with Littlewood began in 1929, with his work towards a fellowship at Trinity, and Hardy writes that "Littlewood's influence dominates nearly all his earliest work". Their work became the foundation for Littlewood–Paley theory, an application of real-variable techniques in complex analysis.[a]
- The Walsh–Paley numeration, a standard method for indexing the Walsh functions, came from a 1932 suggestion of Paley.[b]
- Paley collaborated with Antoni Zygmund on Fourier series, continuing the work on this topic that he had already done with Littlewood. His work in this area also led to the Paley–Zygmund inequality in probability theory.[c]
- In a 1933 paper, he published the Paley construction for Hadamard matrices.[d] In the same paper, he first formulated the Hadamard conjecture on the sizes of matrices for which Hadamard matrices exist. The Paley graphs and Paley tournaments in graph theory are closely related, although they do not appear explicitly in this work. In the context of compressed sensing, frames (partial bases of Hilbert spaces) derived from this construction have been called "Paley equiangular tight frames".
- His collaboration with Norbert Wiener included the Paley–Wiener theorem in harmonic analysis. Paley was originally selected as the 1934 American Mathematical Society Colloquium Lecturer; after his death, Wiener replaced him as speaker, and spoke on their joint work, which was published as a book.[e]
For the short span of his research career, Paley was very productive; Hardy lists 26 of Paley's publications, and more were published posthumously. These publications include:
Littlewood, J. E.; Paley, R. E. A. C. (1931), "Theorems on Fourier Series and Power Series", The Journal of the London Mathematical Society, 6 (3): 230–233, doi:10.1112/jlms/s1-6.3.230, MR 1574750; Littlewood, J. E.; Paley, R. E. A. C. (1936), "Theorems on Fourier series and power series (II)", Proceedings of the London Mathematical Society, Second Series, 42 (1): 52–89, doi:10.1112/plms/s2-42.1.52, MR 1577045; Littlewood, J. E.; Paley, R. E. A. C. (1937), "Theorems on Fourier Series and Power Series(III)", Proceedings of the London Mathematical Society, Second Series, 43 (2): 105–126, doi:10.1112/plms/s2-43.2.105, MR 1575588
- Jones, Gareth A. (2020), "Paley and the Paley graphs", in Jones, Gareth A.; Ponomarenko, Ilia; Širáň, Jozef (eds.), WAGT: International workshop on Isomorphisms, Symmetry and Computations in Algebraic Graph Theory, Pilsen, Czech Republic, October 3–7, 2016, Springer Proceedings in Mathematics & Statistics, vol. 305, Springer, pp. 155–183, doi:10.1007/978-3-030-32808-5_5, S2CID 119129954
- O'Connor, John J.; Robertson, Edmund F., "Raymond Paley", MacTutor History of Mathematics Archive, University of St Andrews
- Hardy, G. H. (1934), "Raymond Edward Alan Christopher Paley", Journal of the London Mathematical Society, 9 (1): 76–80, doi:10.1112/jlms/s1-9.1.76, MR 1574718
- "Mr. R. E. A. C. Paley", The Times, April 1933 – via MacTutor History of Mathematics Archive
- Atiyah, Michael Francis (November 1996), "John Arthur Todd, 23 August 1908 – 22 December 1994", Biographical Memoirs of Fellows of the Royal Society, 42: 483–494, doi:10.1098/rsbm.1996.0029
- Rice, Adrian C.; Wilson, Robin J. (2003), "The rise of British analysis in the early 20th century: the role of G. H. Hardy and the London Mathematical Society", Historia Mathematica, 30 (2): 173–194, doi:10.1016/S0315-0860(03)00002-8, MR 1994357
- Wiener, Norbert (1933), "R. E. A. C. Paley—In memoriam", Bulletin of the American Mathematical Society, 39 (7): 476, doi:10.1090/S0002-9904-1933-05637-9, MR 1562651
- Stein, Elias M. (1970), Topics in harmonic analysis related to the Littlewood–Paley theory, Annals of Mathematics Studies, vol. 63, University of Tokyo Press, MR 0252961
- Frazier, Michael; Jawerth, Björn; Weiss, Guido (1991), Littlewood–Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics, vol. 79, American Mathematical Society, doi:10.1090/cbms/079, ISBN 0-8218-0731-5, MR 1107300
- Trakhtman, V. A. (1973), "Factorization of matrices of the Walsh function ordered according to Paley and repetition frequency", Radiotehn. I Èlektron., 18: 2521–2528, MR 0403781
- Ghosh, B. K. (2002), "Probability inequalities related to Markov's theorem", The American Statistician, 56 (3): 186–190, doi:10.1198/000313002119, JSTOR 3087296, MR 1940206, S2CID 120451773
- MacWilliams, F. J.; Sloane, N. J. A. (1977), The Theory of Error-Correcting Codes, North-Holland, p. 47 and 56, ISBN 0-444-85009-0, MR 0465509
- Hedayat, A.; Wallis, W. D. (1978), "Hadamard matrices and their applications", Annals of Statistics, 6 (6): 1184–1238, doi:10.1214/aos/1176344370, JSTOR 2958712, MR 0523759
- Dustin G. Mixon (June 2012), "The Paley equiangular tight frame as an RIP candidate", Sparse Signal Processing with Frame Theory (PhD thesis), Princeton University, pp. 72–76, arXiv:1204.5958
- Renes, Joseph M. (2007), "Equiangular tight frames from Paley tournaments", Linear Algebra and Its Applications, 426 (2–3): 497–501, arXiv:math/0408287, doi:10.1016/j.laa.2007.05.029, MR 2350673
- Rudin, Walter (1987), "Two theorems of Paley and Wiener", Real and complex analysis (3rd ed.), McGraw-Hill, pp. 372–376, ISBN 0-07-054234-1, MR 0924157