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

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 $p, 1 < p < ∞$.