Multiply universal holomorphic functions
Journal of Approximation Theory
Universal overconvergence of polynomial expansions of harmonic functions
Journal of Approximation Theory
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For a holomorphic function f in the unit disk, S"n(f) denotes the n-th partial sum of the Taylor development of f with center at 0. We show that given a strictly increasing sequence of positive integers (@l"n), there exists a holomorphic function f on the unit disk such that the pairs of partial sums {(S"n(f),S"@l"""n(f)):n=1,2,...} approximate all plausibly approximable functions uniformly on suitable compact subsets K of the complex plane if and only if lim sup"n@l"nn=+~. This provides a new strong notion of universality for Taylor series.