Raymond Edward Alan Christopher Paley (7 January 1907 – 7 April 1933) was an

English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

who made significant contributions to

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...

before dying young in a skiing accident.

Life

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 Eton College () is a Public school (United Kingdom), public school in Eton, Berkshire, England. It was founded in 1440 by Henry VI of England, Henry VI under the name ''Kynge's College of Our Ladye of Eton besyde Windesore'',Nevill, p. 3 ff. i ...

as a King's Scholar and at

Trinity College, Cambridge Trinity College is a constituent college of the University of Cambridge. Founded in 1546 by King Henry VIII, Trinity is one of the largest Cambridge colleges, with the largest financial endowment of any college at either Cambridge or Oxford. ...

. He became a wrangler in 1928, and with J. A. Todd, he was one of two winners of the 1930

Smith's Prize The Smith's Prize was the name of each of two prizes awarded annually to two research students in mathematics and theoretical physics at the University of Cambridge from 1769. Following the reorganization in 1998, they are now awarded under the n ...

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 Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher ...

at the

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of th ...

and with George Pólya at

Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...

, 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. He was killed on 7 April 1933 in a skiing trip to the

Canadian Rockies The Canadian Rockies (french: Rocheuses canadiennes) or Canadian Rocky Mountains, comprising both the Alberta Rockies and the British Columbian Rockies, is the Canadian segment of the North American Rocky Mountains. It is the easternmost part ...

, by an

avalanche An avalanche is a rapid flow of snow down a slope, such as a hill or mountain. Avalanches can be set off spontaneously, by such factors as increased precipitation or snowpack weakening, or by external means such as humans, animals, and ea ...

on Deception Pass.

Contributions

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 nearty all his earliest work". Their work became the foundation for

Littlewood–Paley theory In harmonic analysis, a field within mathematics, Littlewood–Paley theory is a theoretical framework used to extend certain results about ''L''2 functions to ''L'p'' functions for 1 1, then the sequence ''S'n'j'' converges alm ...

, an application of real-variable techniques in

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

. *The Walsh–Paley numeration, a standard method for indexing the

Walsh function In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be used to represent any continuous ...

s, came from a 1932 suggestion of Paley. *Paley collaborated with

Antoni Zygmund Antoni Zygmund (December 25, 1900 – May 30, 1992) was a Polish mathematician. He worked mostly in the area of mathematical analysis, including especially harmonic analysis, and he is considered one of the greatest analysts of the 20th century. ...

Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...

, 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 Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

. *In a 1933 paper, he published the

Paley construction In mathematics, the Paley construction is a method for constructing Hadamard matrix, Hadamard matrices using finite fields. The construction was described in 1933 by the England, English mathematician Raymond Paley. The Paley construction uses qua ...

for

Hadamard matrices In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in ...

. In the same paper, he first formulated the

Hadamard conjecture In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows ...

on the sizes of matrices for which Hadamard matrices exist. The

Paley graph In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, ...

s and Paley

tournaments A tournament is a competition involving at least three competitors, all participating in a sport or game. More specifically, the term may be used in either of two overlapping senses: # One or more competitions held at a single venue and concentr ...

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

are closely related, although they do not appear explicitly in this work. In the context of

compressed sensing Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a Signal (electronics), signal, by finding solutions to Underdetermined ...

, frames (partial bases of

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

s) derived from this construction have been called "Paley equiangular tight frames". *His collaboration with

included the

Paley–Wiener theorem In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (189 ...

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...

. Paley was originally selected as the 1934

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

Colloquium Lecturer; after his death, Wiener replaced him as speaker, and spoke on their joint work, which was published as a book.

Selected publications

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:

References

{{DEFAULTSORT:Paley, Raymond 1907 births 1933 deaths Scientists from Bournemouth 20th-century English mathematicians Graph theorists Mathematical analysts People educated at Eton College Alumni of Trinity College, Cambridge Fellows of Trinity College, Cambridge Natural disaster deaths in Canada Deaths in avalanches