Gábor Szegő (Hungarian: [ˈɡaːbor ˈsɛɡøː]) (January 20, 1895 – August 7, 1985) was a Hungarian mathematician. He was one of the foremost analysts of his generation and made fundamental contributions to the theory of Toeplitz matrices and orthogonal polynomials.
Szegő was born in Kunhegyes, Austria-Hungary (today Hungary), into a Jewish family as the son of Adolf Szegő and Hermina Neuman. He married the chemist Anna Elisabeth Neményi in 1919, with whom he had two children.
In 1912 he started studies in mathematical physics at the University of Budapest, with summer visits to the University of Berlin and the University of Göttingen, where he attended lectures by Frobenius and Hilbert, amongst others. In Budapest he was taught mainly by Fejér, Beke, Kürschák and Bauer and made the acquaintance of his future collaborators George Pólya and Michael Fekete. His studies were interrupted in 1915 by World War I, in which he served in the infantry, artillery and air corps. In 1918 while stationed in Vienna, he was awarded a doctorate by the University of Vienna for his work on Toeplitz determinants. He received his Privat-Dozent from the University of Berlin in 1921, where he stayed until being appointed as successor to Knopp at the University of Königsberg in 1926. Intolerable working conditions during the Nazi regime resulted in a temporary position at the Washington University in Saint Louis, Missouri in 1936, before his appointment as chairman of the mathematics department at Stanford University in 1938, where he helped build up the department until his retirement in 1966. He died in Palo Alto, California. His doctoral students include Paul Rosenbloom and Joseph Ullman.
Szegő's most important work was in analysis. He was one of the foremost analysts of his generation and made fundamental contributions to the theory of Toeplitz matrices and orthogonal polynomials. He wrote over 130 papers in several languages. Each of his four books, several written in collaboration with others, has become a classic in its field. The monograph Orthogonal polynomials, published in 1939, contains much of his research and has had a profound influence in many areas of applied mathematics, including theoretical physics, stochastic processes and numerical analysis.
Tutoring von Neumann
At the age of 15, the young John von Neumann, recognised as a mathematical prodigy, was sent to study advanced calculus under Szegő. On their first meeting, Szegő was so astounded by von Neumann's mathematical talent and speed that he was brought to tears. Szegő subsequently visited the von Neumann house twice a week to tutor the child prodigy. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out with his father's stationery, are now on display at the von Neumann archive at Budapest.
Amongst the many honours received during his lifetime were:
- Julius König Prize of the Hungarian Mathematical Society (1928)
- Member of the Königsberger Gelehrten Gesellschaft (1928)
- Corresponding member of the Austrian Academy of Sciences in Vienna (1960)
- Honorary member of the Hungarian Academy of Sciences (1965)
- Fekete–Szegő inequality
- Grace–Walsh–Szegő coincidence theorem
- Rogers–Szegő polynomials
- Szegő inequality
- Szegő kernel
- Szegő limit theorems
- Szegő polynomial
- Problems and theorems in analysis
- The collected Papers of Gábor Szegő, 3 Vols (ed. Richard Askey), Birkhäuser, 1982, ISBN 3-7643-3063-5
- Pólya, George; Szegő, Gábor (1925, 1972), Problems and theorems in analysis, 2 Vols, Springer-Verlag
- Szegő, Gábor (1933), Asymptotische Entwicklungen der Jacobischen Polynome, Niemeyer
- Szegő, Gábor (1939, 1955), Orthogonal Polynomials, American Mathematical Society 
- Pólya, George; Szegő, Gábor (1951), Isoperimetric problems in mathematical physics, Annals of Mathematics Studies 27, Princeton University Press
- Szegő, Gábor; Grenander, Ulf (1958), Toeplitz forms and their applications, Chelsea
- Biography on the homepage of Kunhegyes (in Hungarian)
- Biographies - Alfred Rényi Institute of Mathematics (bios of Bauer Mihály and Beke Manó)
- Gábor Szegő at the Mathematics Genealogy Project
- Official memorial citation, Stanford University.
- Impagliazzo, John; Glimm, James; Singer, Isadore Manuel The Legacy of John von Neumann, American Mathematical Society, 1990, ISBN 0-8218-4219-6.
- John Von Neumann: The Scientific Genius Who Pioneered the Modern Computer, by Norman Macrae, American Mathematical Soc., 2000, page 70
- Shohat, J. (1935). "Szegő on Jacobi Polynomials". Bull. Amer. Math. Soc. 41 (3): 165–169.
- Shohat, J. (1940). "Review: Gabor Szegő: Orthogonal Polynomials". Bull. Amer. Math. Soc. 46 (7): 583–587. doi:10.1090/s0002-9904-1940-07231-3.
- Spitzer, F. (1959). "Review: Ulf Grenander and Gabor Szegő, Toeplitz forms and their applications". Bull. Amer. Math. Soc. 65 (2): 97–101. doi:10.1090/s0002-9904-1959-10296-2.
- O'Connor, John J.; Robertson, Edmund F., "Gábor Szegő", MacTutor History of Mathematics archive, University of St Andrews.
- Askey, Richard (17 May 1995). "Gábor Szegő - One hundred years (Topic #25)". OP-SF NET.