ESTIMATES OF FUNCTIONALS BY GENERALIZED MODULI OF CONTINUITY GENERATED BY THE DUNKL OPERATORS

摘要

We introduce moduli of continuity on the basis of the generalized translation generated by the Dunkl operatorsrnΛf(x)＝f'(x)+(2α+1)/x (f(x)-f(-x))/2rnand estimate the functionals in terms of these moduli. The constants in these inequalities are explicitly expressed in terms of the moments of functionals, i.e. their upper bounds on generalized Sobolev classes. Jackson-type inequalities in the Lebesgue spaces on the real line with the weight A_α(x) = |x|~(2α+1), α > -1/2, are particular cases of the inequalities established.