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Title:聽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 $f_r, r < 1$. (Joint work with Javad Mashreghi and Pierre-Olivier Parise).

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