2017
Том 69
№ 6

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Estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of $L_p$

Konopovich T. O.

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Abstract

We obtain upper bounds in terms of Fourier coefficients for the best approximation by an “angle” and for norms in the metric of $L_p$ for functions of two variables defined by trigonometric series with coefficients such that $a_{l_1 l_2} → 0$ as $l_1 + l_2 → ∞$ and $$\mathop \sum \limits_{k_1 = 0}^\infty \mathop \sum \limits_{k_2 = 0}^\infty \left( {\mathop \sum \limits_{l_1 = k_1 }^\infty \mathop \sum \limits_{l_2 = k_2 }^\infty \left| {\Delta ^{12} a_{l_1 \;l_2 } } \right|} \right)^p (k_1 + 1)^{p - 2} \;(k_2 + 1)^{p - 2} < \infty$$ for a certain $p, 1 < p < ∞$.

English version (Springer): Ukrainian Mathematical Journal 56 (2004), no. 9, pp 1403–1416.

Citation Example: Konopovich T. O. Estimation of the best approximation of periodic functions of two variables by an “angle” in the metric of $L_p$ // Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1182–1192.

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