Raymond Edward Alan Christopher Paley (7 January 1907 – 7 April 1933) was an Englishmathematician who made significant contributions to mathematical analysis before dying young in a skiing accident.
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.
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".[3] Their work became the foundation for Littlewood–Paley theory, an application of real-variable techniques in complex analysis.[8][9][a]
The Walsh–Paley numeration, a standard method for indexing the Walsh functions, came from a 1932 suggestion of Paley.[10][b]
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:
a.
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, MR1574750; 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, MR1577045; 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, MR1575588
b.
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, MR1576148, Zbl0005.24806
Paley, R. E. A. C. (1933), "On orthogonal matrices", Journal of Mathematics and Physics, 12 (1–4), Massachusetts Institute of Technology: 311–320, doi:10.1002/sapm1933121311, Zbl0007.10004
e.
Paley, Raymond E. A. C.; Wiener, Norbert (1934), Fourier Transforms in the Complex Domain, Colloquium Publications, vol. 19, Providence, Rhode Island: American Mathematical Society, Zbl0011.01601
References
^ abcJones, 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, ISBN978-3-030-32807-8, S2CID119129954
^Stein, Elias M. (1970), Topics in harmonic analysis related to the Littlewood–Paley theory, Annals of Mathematics Studies, vol. 63, University of Tokyo Press, MR0252961
^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, ISBN0-8218-0731-5, MR1107300
^Trakhtman, V. A. (1973), "Factorization of matrices of the Walsh function ordered according to Paley and repetition frequency", Radiotehn. I Èlektron., 18: 2521–2528, MR0403781
^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