Contributions of Issai Schur to Analysis

TL;DR

This survey comprehensively reviews Issai Schur's extensive contributions to analysis, covering topics like the Schur test, convexity, complements, and interpolation, highlighting their significance and generalizations across mathematics and engineering.

Contribution

The paper provides a detailed survey of Schur's work in analysis, including lesser-known topics and their influence on various mathematical fields.

Findings

Schur's work on the Schur test and multipliers is fundamental in estimate theory.

Schur's contributions to convexity and complements have broad applications.

The survey highlights the influence of Schur's analysis on modern mathematical research.

Abstract

The name Schur is associated with many terms and concepts that are widely used in a number of diverse fields of mathematics and engineering. This survey article focuses on Schur's work in analysis. Here too, Schur's name is commonplace: The Schur test and Schur-Hadamard multipliers (in the study of estimates for Hermitian forms), Schur convexity, Schur complements, Schur's results in summation theory for sequences (in particular, the fundamental Kojima-Schur theorem), the Schur-Cohn test, the Schur algorithm, Schur parameters and the Schur interpolation problem for functions that are holomorphic and bounded by one in the unit disk. In this survey, we discuss all of the above mentioned topics and then some, as well as some of the generalizations that they inspired. There are nine sections of text, each of which is devoted to a separate theme based on Schur's work. Each of these sections has an independent bibliography. There is very little overlap. A tenth section presents a list of the papers of Schur that focus on topics that are commonly considered to be analysis. We begin with a review of Schur's less familiar papers on the theory of commuting differential operators.