Power series as Fourier series

TL;DR

This paper develops an abstract theory of Fourier series in locally convex topological vector spaces, proves an analog of Fejér's theorem, and applies it to distributional solutions of complex analysis, connecting classical results to series expansions.

Contribution

It introduces a generalized Fourier series framework in locally convex spaces and demonstrates its applications to complex analysis and distribution theory.

Findings

Proved an analog of Fejér's theorem for these series

Applied the theory to distributional solutions of Cauchy-Riemann equations

Derived classical function theory results from series expansions

Abstract

An abstract theory of Fourier series in locally convex topological vector spaces is developed. An analog of Fej\'{e}r's theorem is proved for these series. The theory is applied to distributional solutions of Cauchy-Riemann equations to recover basic results of complex analysis. Some classical results of function theory are also shown to be consequences of the series expansion.