Perfect spline approximation |
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Authors: | Y -K Hu X M Yu |
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Institution: | a Department of Mathematics and Computer Science, Georgia Southern University, Statesboro, GA 30460-8093, USA;b Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804, USA |
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Abstract: | Our study of perfect spline approximation reveals: (i) it is closely related to ΣΔ modulation used in one-bit quantization of bandlimited signals. In fact, they share the same recursive formulae, although in different contexts; (ii) the best rate of approximation by perfect splines of order r with equidistant knots of mesh size h is hr−1. This rate is optimal in the sense that a function can be approximated with a better rate if and only if it is a polynomial of degree <r.The uniqueness of best approximation is studied, too. Along the way, we also give a result on an extremal problem, that is, among all perfect splines with integer knots on
, (multiples of) Euler splines have the smallest possible norms. |
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Keywords: | Perfect splines Spline approximation Sigma– Delta modulation |
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