Failure of Approximation of Odd Functions by Odd Polynomials

TL;DR

This paper constructs a specific Hilbert space of holomorphic functions where odd polynomials are not dense among odd functions, demonstrating limitations in approximation methods for certain functions.

Contribution

It introduces a Hilbert space where polynomial density does not imply odd polynomial density, revealing new limitations in function approximation.

Findings

Existence of a function outside the span of Taylor partial sums

Existence of a function outside the span of radial dilates

Demonstration of failure of approximation by certain methods

Abstract

We construct a Hilbert holomorphic function space $H$ on the unit disk such that the polynomials are dense in $H$, but the odd polynomials are not dense in the odd functions in $H$. As a consequence, there exists a function $f$ in $H$ that lies outside the closed linear span of its Taylor partial sums $s_n(f)$, so it cannot be approximated by any triangular summability method applied to the $s_n(f)$. We also show that there exists a function $f$ in $H$ that lies outside the closed linear span of its radial dilates $f_r, ~r<1$.