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Moduli of smoothness and best approximation of functions with singularities

✍ Scribed by M. Ganzburg

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
621 KB
Volume
40
Category
Article
ISSN
0898-1221

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✦ Synopsis

Error estimates for approximation of functions ~ox,a,0 (x) = ~x,~,l (x) + i~ox,a,2 (X) = [x[)~exp (iA[x[-Β°'), A > 0, a :> 0, A e R are given. Let E(f,B, Lp(f~)) denote the error of approximation of f by elements from B in the Lp-metric. Then, it is shown that for polynomial approximation E(~,a,l,Pr,,Lp(-a,a)) ~' n -(;~+l/p)/(l+Β°') holds true for 1 < p ~ oo, 0 < i < 2.

The similar estimates are also valid for the errors of approximation by entire functions of exponential type, trigonometric polynomials, and periodic splines. The proofs are based on exact estimates of the moduli of smoothness of qa~,a,i and a general Stechkin-type theorem. (~) 2000 Elsevier Science Ltd. All rights reserved.

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