Failure of Approximation of Odd Functions by Odd Polynomials

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 \(f_r, ~r<1\).