Raymond Paley

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Raymond E. A. C. Paley
Born(1907-01-07)7 January 1907
Died7 April 1933(1933-04-07) (aged 26)
Deception Pass, Fossil Mountain, in the Canadian Rockies
Alma materUniversity of Cambridge
Known for
AwardsSmith's Prize (1930)
Scientific career
Photograph of Paley's grave in The Old Banff Cemetery. The main gate is visible on the left.

Raymond Edward Alan Christopher Paley (7 January 1907 – 7 April 1933) was an English mathematician who made significant contributions to mathematical analysis before dying young in a skiing accident.


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[1] and at Trinity College, Cambridge.[2] He became a wrangler in 1928,[3] and with J. A. Todd, he was one of two winners of the 1930 Smith's Prize examination.[2][3]

He was elected a Research Fellow of Trinity College in 1930,[4] edging out Todd for the position,[5] 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.[6] 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,[1] 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.[7]

He was killed on 7 April 1933 in a skiing trip to the Canadian Rockies, by an avalanche on Deception Pass.[2]

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.

Selected publications[edit]

For the short span of his research career, Paley was very productive; Hardy lists 26 of Paley's publications,[3] 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
Paley, R. E. A. C. (1932), "A Remarkable Series of Orthogonal Functions I, II", Proceedings of the London Mathematical Society, Second Series, 34 (4): 241–264, 265–279, doi:10.1112/plms/s2-34.1.241, MR 1576148, Zbl 0005.24806
Paley, R. E. A. C.; Zygmund, Antoni (1932), "A note on analytic functions in the unit circle", Proceedings of the Cambridge Philosophical Society, 28 (3): 266–272, Bibcode:1932PCPS...28..266P, doi:10.1017/S0305004100010112, Zbl 0005.06602
Paley, R. E. A. C. (1933), "On orthogonal matrices", Journal of Mathematics and Physics, Massachusetts Institute of Technology, 12 (1–4): 311–320, doi:10.1002/sapm1933121311, Zbl 0007.10004
Paley, Raymond E. A. C.; Wiener, Norbert (1934), Fourier Transforms in the Complex Domain, Colloquium Publications, vol. 19, Providence, Rhode Island: American Mathematical Society, Zbl 0011.01601


  1. ^ a b c 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
  2. ^ a b c d e O'Connor, John J.; Robertson, Edmund F., "Raymond Paley", MacTutor History of Mathematics Archive, University of St Andrews
  3. ^ a b c d 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
  4. ^ "Mr. R. E. A. C. Paley", The Times, April 1933 – via MacTutor History of Mathematics Archive
  5. ^ 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
  6. ^ 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
  7. ^ a b 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
  8. ^ 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
  9. ^ 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
  10. ^ 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
  11. ^ 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
  12. ^ 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
  13. ^ 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
  14. ^ 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
  15. ^ 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
  16. ^ 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