Best One-Sided L 1-Approximation by Blending Functions of Order (2, 2)
✍ Scribed by Dimiter Dryanov; Petar Petrov
- Publisher
- Elsevier Science
- Year
- 2002
- Tongue
- English
- Weight
- 189 KB
- Volume
- 115
- Category
- Article
- ISSN
- 0021-9045
No coin nor oath required. For personal study only.
✦ Synopsis
Let f ¥ C 2, 2 ([ -1, 1] 2 ) be a real function satisfying " 4 f/"x 2 "y 2 \ 0 on [ -1, 1] 2 . We study the problem of best one-sided L 1 -approximation to f from the linear space {h ¥ C 2, 2 ([ -1, 1] 2 ) : " 4 h/"x 2 "y 2 =0} of all blending functions of order (2, 2). The unique best one-sided L 1 -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 L 1 -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 " 4 f/"x 2 "y 2 \ 0 on [ -1, 1] 2 when we approximate from below. The best one-sided L 1 -approximant from above has the smoothness of f. The best one-sided L 1 -approximant to f from below is a blendingspline function with two line segment knots {(x, 0) : -1 [ x [ 1} and {(0, y): -1 [ y [ 1}; i.e., the best one-sided approximation to f from below possesses a saturation effect with respect to the smoothness of f.