# PADS Seminar: Schur functions- A brief survey

**Author: Lona**

**Author: Lona**

Speaker: Jaydeb Sarkar, Indian Statistical Institute

**Abstract: **The Schur class, denoted by S(D), is the set of all functions analytic and bounded by one in modulus in the open unit disc D in the complex plane. Elements of S(D) are called Schur functions. A classical result going back to Issai Schur states: A function f is a Schur function if and only if f admits a linear fractional transformation (or transfer function realization). Linear fractional transformations are attached with colligation matrices or scattering matrices on Hilbert spaces. Schur’s view of bounded analytic functions is one of the most used (and useful) tools in classical and modern complex analysis, function theory, operator theory, electrical network theory, signal processing, linear systems, operator algebras, and image processing (just to name a few).

In the first part of this talk, we will give a brief (but within the span of little more than a century) historic perspective and introduction to Schur theory and discuss its interactions with some classical problems in function theory and operator theory (like Nevanlinna-Pick interpolation). In the second part of the talk, we will review Schur’s approach (ubiquity and its complications) to functions of several complex variables from a linear analysis point of view.