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Izv. RAN. Ser. Mat., 2015, Volume 79, Issue 2, Pages 45–76 (Mi izv8094)  

Estimates for the error of approximation of functions in $L_p^1$ by polynomials and partial sums of series in the Haar and Faber–Schauder systems

S. B. Vakarchuka, A. N. Shchitovb

a Dnepropetrovsk University of Economics and Law
b Ukrainian Academy of Customs, Dnipropetrovsk

Abstract: We find exact estimates for the error of approximation of functions in the classes $L_p^1$ by polynomials in the Haar system and partial sums of the Faber–Schauder series in the metrics of the spaces $L_p$. The error in approximating a function $f\in L_p^1$ is estimated in terms of the norms of the first derivatives $\|f^{(1)}\|_{L_p}$ and $\|f^{(1)}-\overline S^{(1)}_n(f)\|_{L_p}$. The resulting bounds are unimprovable for some values of $n$.

Keywords: Haar system of functions, Faber–Schauder system of functions, best approximation of functions by polynomials, one-sided approximation of functions by polynomials.

DOI: https://doi.org/10.4213/im8094

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English version:
Izvestiya: Mathematics, 2015, 79:2, 257–287

Bibliographic databases:

UDC: 517.5
MSC: 41A25
Received: 21.01.2013
Revised: 31.07.2014

Citation: S. B. Vakarchuk, A. N. Shchitov, “Estimates for the error of approximation of functions in $L_p^1$ by polynomials and partial sums of series in the Haar and Faber–Schauder systems”, Izv. RAN. Ser. Mat., 79:2 (2015), 45–76; Izv. Math., 79:2 (2015), 257–287

Citation in format AMSBIB
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