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Mat. Sb., 2010, Volume 201, Number 11, Pages 137–160 (Mi msb7586)  

Approximating smooth functions using algebraic-trigonometric polynomials

I. I. Sharapudinov

Daghestan Scientific Centre of the Russian Academy of Sciences

Abstract: The problem under consideration is that of approximating classes of smooth functions by algebraic-trigonometric polynomials of the form $p_n(t)+\tau_m(t)$, where $p_n(t)$ is an algebraic polynomial of degree $n$ and $\tau_m(t)=a_0+\sum_{k=1}^ma_k\cos k\pi t+b_k\sin k\pi t$ is a trigonometric polynomial of order $m$. The precise order of approximation by such polynomials in the classes $W^r_\infty(M)$ and an upper bound for similar approximations in the class $W^r_p(M)$ with $\frac43<p<4$ are found. The proof of these estimates uses mixed series in Legendre polynomials which the author has introduced and investigated previously.
Bibliography: 13 titles.

Keywords: classes of smooth functions, algebraic-trigonometric polynomials, simultaneous approximation of functions and derivatives, mixed series in Legendre polynomials.

DOI: https://doi.org/10.4213/sm7586

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English version:
Sbornik: Mathematics, 2010, 201:11, 1689–1713

Bibliographic databases:

UDC: 517.518
MSC: Primary 41A30; Secondary 41A10, 42A10
Received: 09.06.2009 and 06.05.2010

Citation: I. I. Sharapudinov, “Approximating smooth functions using algebraic-trigonometric polynomials”, Mat. Sb., 201:11 (2010), 137–160; Sb. Math., 201:11 (2010), 1689–1713

Citation in format AMSBIB
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