Optimal Approximation by sk-Splines on the Torus

2018/04/09 by Oliveira, Juliana Gaiba, Tozoni, Sergio Antonio
#41A15 #41A25 #41A65 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1804.03106

Abstract

Fixed a continuous kernel K on the d-dimensional torus, we consider a generalization of the univariate sk-spline to the torus, associated with the kernel K. It is proved an estimate which provides the rate of convergence of a given function by its interpolating sk-splines, in the norm of L^q for functions of the type f=K*φ where φ∈ L^p and 1≤ p ≤ 2 ≤ q ≤ ∞, 1/p - 1/q ≥ 1/2. The rate of convergence is obtained for functions f in Sobolev classes and this rate gives optimal error estimate of the same order as best trigonometric approximation, in a special case.

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