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arXiv · 2006.16871

Failure of Approximation of Odd Functions by Odd Polynomials

Javad Mashreghi·Pierre-Olivier Parisé·Thomas Ransford
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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 $f_r, ~r<1$.

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Javad Mashreghi, Pierre-Olivier Parisé, Thomas Ransford. 2020-06-30. Failure of Approximation of Odd Functions by Odd Polynomials. https://arxiv.org/abs/2006.16871

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