The degree of approximation by polynomials on some disjoint intervals in the complex plane
- Hasson, Maurice
Publication date
January 2007
DOI Publisher
Elsevier Inc. ISSN
0021-9045 Citation count (estimate)
4Abstract
AbstractLet f(z) be a continuous function defined on the compact set K⊂C and let En(f)=En(f,K) be the degree of approximation to f, for the supremum norm on K, by polynomials of degree (at most) n. ThusEn(f,K)=infP∈Pn∥f-P∥.Here Pn denotes the space of polynomials of degree at most n and ∥.∥ is the supremum norm on K.For a positive integer s and for 0<a<b, letK=Ka,bs=⋃k=0s-1e2πiks[a,b].We show that for a large class of piecewise analytic functions f defined on Klimsupn→∞En(f,K)1n=bs2-as2bs2+as2s,thus recovering several classical results.The proof of this error estimate is then translated into an algorithm that finds the polynomial of near best approximation
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