# Matrix transformations of series of orthogonal polynomials

Borwein, David and Borwein, Peter and Jakimovski, Amnon (1995) Matrix transformations of series of orthogonal polynomials. [Preprint]

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For a sequence of polynomials $(P_n)$ orthonormal on the interval $[-1,1]$, we consider the sequence of transforms $(g_ n)$ of the series $\sum_{k=0}^{\infty}a_kP_k(u)$ given by $g_n(u):=\text{$\sum_{k=0}^{\infty}b_{nk}a_kP_k(u)$}$. We establish necessary and sufficient conditions on the matrix $(b_{nk})$ for the sequence $(g_n)$ to converge uniformly on compact subsets of the interior of an appropriate ellipse to a function holomorphic on that interior.