Представление и приб лижение периодических функц ий многих переменных с заданным смешанным модулем непрепывнос ти

Abstract

In this paper the author studies classesH ^Ω _q of periodic functions of several variables whose mixed moduli of continuity do not exceed a given modulus of continuity ω(t ₁ ...,t _d ). Necessary and sufficient conditions of belonging of a functionf(x ₁, ...,x _d ) to the classH ^Ω _q are considered (Theorem 1). These necessary and sufficient conditions are proved under some additional assumptions on ω(t ₁, ...,t _d ). It is shown that additional assumptions cannot be omitted (Theorem 3). Besides, the estimates of best approximations of classesH ^Ω _q with some special ω(t ₁, ...,t _d ) are given (Theorems 4 and 5).