In Budapest he was taught mainly by Fejér, Beke, Kürschák and Bauer[2] and made the acquaintance of his future collaborators George Pólya and Michael Fekete.

He was one of the foremost analysts of his generation and made fundamental contributions to the theory of Toeplitz matrices and orthogonal polynomials.

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.

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, as recalled by his wife, he came back home with tears in his eyes.