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: Culture, language and peoples * ''English'', an adjective for something of, from, or related to England * ''English'', an Amish ter ...

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, mathematical structure, structure, space, Mathematica ...

who made significant contributions to

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

before dying young in a skiing accident.

Life

Paley was born in

Bournemouth, England Bournemouth ( ) is a coastal resort town in the Bournemouth, Christchurch and Poole unitary authority area, in the ceremonial county of Dorset, England. At the 2021 census, the built-up area had a population of 196,455, making it the largest ...

, 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 providing boarding school, boarding education for boys aged 13–18, in the small town of Eton, Berkshire, Eton, in Berkshire, in the United Kingdom. It has educated Prime Mini ...

as a

King's Scholar A King's Scholar, abbreviated KS in the United Kingdom, is the recipient of a scholarship from a foundation created by, or under the auspices of, a British monarch. The scholarships are awarded at certain Public school (United Kingdom), public ...

and at

Trinity College, Cambridge Trinity College is a Colleges of the University of Cambridge, 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 ...

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

Smith's Prize 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 names ...

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 John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanu ...

. After the 1931 return of

G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...

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 computer scientist, mathematician, and philosopher. He became a professor of mathematics at the Massachusetts Institute of Technology ( MIT). A child prodigy, Wiener late ...

at the

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...

and with

George Pólya George Pólya (; ; December 13, 1887 – September 7, 1985) was a Hungarian-American mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributi ...

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

, and as part of the same trip also planned to work with

Lipót Fejér Lipót Fejér (or Leopold Fejér, ; 9 February 1880 – 15 October 1959) was a Hungarian mathematician of Jewish heritage. Fejér was born Leopold Weisz, and changed to the Hungarian name Fejér around 1900. Biography He was born in Pécs, Au ...

at a seminar in Chicago organized as part of the

Century of Progress A Century of Progress International Exposition, also known as the Chicago World's Fair, was a world's fair held in the city of Chicago, Illinois, United States, from 1933 to 1934. The fair, registered under the Bureau International des Exposit ...

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

Canadian Rockies The Canadian Rockies () 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 of the Canadian Cordillera, w ...

, by an

avalanche An avalanche is a rapid flow of snow down a Grade (slope), slope, such as a hill or mountain. Avalanches can be triggered spontaneously, by factors such as increased precipitation or snowpack weakening, or by external means such as humans, othe ...

Deception Pass Deception Pass (; ) is a strait separating Whidbey Island from Fidalgo Island, in the northwest part of the U.S. state of Washington (state), Washington. It connects Skagit Bay, part of Puget Sound, with the Strait of Juan de Fuca. A pair of bri ...

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 nearly 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 a ...

, 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 functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

. *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 f ...

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

Antoni Zygmund Antoni Zygmund (December 26, 1900 – May 30, 1992) was a Polish-American mathematician. He worked mostly in the area of mathematical analysis, including harmonic analysis, and he is considered one of the greatest analysts of the 20th century. ...

Fourier series A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...

, 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 mathematics, the Paley–Zygmund inequality bounds the probability that a positive random variable is small, in terms of its first two moments. The inequality was proved by Raymond Paley and Antoni Zygmund. Theorem: If ''Z'' ≥ 0 i ...

probability theory Probability theory or probability calculus 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 expre ...

. *In a 1933 paper, he published the Paley construction for

Hadamard matrices In mathematics, an 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 a H ...

. In the same paper, he first formulated the Hadamard conjecture on the sizes of matrices for which Hadamard matrices exist. The

Paley graph In mathematics, Paley graphs are 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, which yiel ...

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 and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

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 s ...

, frames (partial bases of

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

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 a theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. It is named after Raymond Paley (1907–1933) and Norbert Wiener (1894–19 ...

harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...

. 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