Best One-Sided L¹-Approximation by Blending Functions of Order (2, 2)

Authors: Dryanov D.¹; Petrov P.²

Source: Journal of Approximation Theory, Volume 115, Number 1, March 2002 , pp. 72-99(28)

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Abstract:

Let f isin C^2, 2([-1, 1]²) be a real function satisfying part ⁴f/ part x² part y² ges 0 on [-1, 1]². We study the problem of best one-sided L¹-approximation to f from the linear space {h isin C^2, 2([-1, 1]²): part ⁴h/ part x² part y²=0} of all blending functions of order (2, 2). The unique best one-sided L¹-approximant to f from above is characterized by transfinite Hermite interpolation on the canonical grid {(x, y) isin [-1, 1]² : |x|=|y|}. For f even with respect to one of its variables we characterize the unique best one-sided L¹-approximant to f from below by transfinite Hermite interpolation on the canonical grid {(x, y) isin [-1, 1]² : |x|+|y|=1}. There is no canonical grid for the entire cone class of functions f with part ⁴f/ part x² part y² ges 0 on [-1, 1]² when we approximate from below. The best one-sided L¹-approximant from above has the smoothness of f. The best one-sided L¹-approximant to f from below is a blending-spline function with two line segment knots {(x, 0): -1 les x les 1} and {(0, y): -1 les y les 1}; i.e., the best one-sided approximation to f from below possesses a saturation effect with respect to the smoothness of f. © 2002 Elsevier Science (USA).

Keywords: blending functions; multivariate approximation; Markov'; s theorem; canonical point sets; best one-sided L¹-approximation; transfinite Hermite interpolation

Language: English

Document Type: Research article

Affiliations: 1: Département de Mathématiques, Université de Montréal, Pavillon André-Aisenstadt, Montréal, P.Q., H3C 3J7, Canada 2: Department of Mathematics, Sofia University, Blvd. J. Boucher 5, Sofia, 1164, Bulgaria

Publication date: 2002-03-01