A Study of Polynomial Convexity and Polynomial Approximations on Certain Classes of Sets

Mondal, Golam Mostafa (2023) A Study of Polynomial Convexity and Polynomial Approximations on Certain Classes of Sets. PhD thesis, Indian Institute of Science Education and Research, Kolkata.

Text (PhD thesis of Golam Mostafa Mondal (16RS062)) 16RS062.pdf - Submitted Version Restricted to Repository staff only Download (1MB)

Abstract

Everyone who works in mathematics understands how vital it is to be able to reduce a problem via approximation. Polynomials are a beautiful family of functions in analysis. Note that a sequence that converges uniformly has many significant properties, e.g., interchanging limits with summation and integration is possible. As a result, approximating a function with polynomials is a significant simplification. The concept of convexity of compact sets in the complex Euclidean space Cn is fundamental in studying function theory in several complex variables. In particular, polynomial convexity is highly important in the approximation theory. Generally it is difficult to show that a compact set in Cn is polynomially convex. It is also challenging to approximate a continuous function by holomorphic polynomials, even if the compact set is polynomially convex. Therefore, it is crucial to find some criteria or give some new class of compact sets that may help with this task. Several authors have done substantial works for totally real submanifolds or totally real outside a small set of points. Still, there are many unanswered questions in this direction. This thesis mainly focuses on polynomial convexity and polynomial approximation of compact subsets of Cn in several different settings. In some cases, the presence of an analytic disc is the only obstruction to polynomial approximation. We also present some such classes of compacts in this thesis.

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