Formula omitted]-widths of certain function classes defined by the modulus of continuity

Abstract

Several exact inequalities are found between the best approximation by trigonometric polynomials in L2 of differentiable periodic functions, in the sense of Weyl, averaged with the weight of the modulus of continuity of arbitrary fractional order. For some classes of functions, defined by specific moduli of continuity, the exact values of the various n-widths in L2 are calculated. In particular the problem of minimizing the constants in inequalities of Jackson–Stechkin type over all subspaces of dimension N is solved. It is proved that this value is equal to the exact value of the different widths of the class L2(α)(β,p,h,φ), where φ is a non-negative integrable weight function on (0,h](0<h≤π/n).