Approximation of periodic functions of several variables with bounded mixed difference

This paper studies questions concerning the approximation of functions of several variables by trigonometric polynomials whose harmonics lie in a “hyperbolic cross” and also properties of functions which do not have harmonics lying in a “hyperbolic cross”. Analogues of H. Bohr's inequality are obtained for such functions. Estimates of optimal order are obtained for the upper bounds of best approximations of certain classes of functions, defined using mixed differences, by trigonometric polynomials whose harmonics lie in a “hyperbolic cross”. The diameters of certain classes are found.
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