An extremum problem on a class of differentiable functions of several variables

Abstract

On the multidimensional classW ^r₀ H ⁽ⁿ⁾_ω of continuous periodic functionsF with therth derivativeD ^r F from

$$H_\omega ^{(n)} = \left\{ {f \in C| |f(x) - f(y)| \leqslant \sum\limits_{i = 1}^n {\omega _i } (|x_i - y_i |)\forall x, y \in \mathbb{R}^n } \right\}$$

(where the ω _i (x _i ) are the convex moduli of continuity) and zero mean with respect to each variable, we obtain the exact value of

$$M_r (\omega ) = \mathop {\sup }\limits_{F \in W_0^r H_\omega ^{(n)} } \left\| F \right\|c$$

.