Best One-Sided L 1-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)
Publisher: Academic Press
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Abstract:
Let f
C2, 2([-1, 1]2) be a real function satisfying 
4f/x2 y2
0 on [-1, 1]2. We study the problem of best one-sided L1-approximation to f from the linear space {hC2, 2([-1, 1]2): 4h/x2 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)[-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)[-1, 1]2 : |x|+|y|=1}. There is no canonical grid for the entire cone class of functions f with 4f/x2 y20 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
x1} and {(0, y): -1y1}; 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
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