Roots under convolution of sequences |
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Authors: | J Korevaar |
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Institution: | Department of Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TVAmsterdam, the Netherlands |
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Abstract: | The convolution a * b of the sequences a = a0, a1, a2, and b is the sequence with elements ∑0n akbn − k. One sets 1, 1, 1, equal to σ. Given that a * a with a ≥ 0 is close to σ * σ, how close is a to σ? More generally, one asks how close a is to σ if the p-th convolution power, a*P with a ≥ 0, is close to σ*P. Power series and complex analysis form a natural tool to estimate the ‘summed deviation’ ρ = σ * (a — σ) in terms of b = a * a — σ * σ or b = a*P − σ*P. Optimal estimates are found under the condition ∑k=0n bk2 = %plane1D;512;(n2β + 1) whenever −½ < β < p − 1. It is not known what the optimal estimates are for the special case bn = %plane1D;512;(nβ). |
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