Characterization of Hölder spaces corresponding to Ditzian-Totik modulus of smoothness

Abstract

For the step-weight function\(\varphi \left( x \right) = \sqrt {1 - x^2 } \), we prove that the Holder spaces \gL{ina, p} on the interval [−1, 1], defined in terms of moduli of smoothness with the step-weight function ϕ, are linearly isomorphic to some sequence spaces, and the isomorphism is given by the coefficients of function with respect to a system of orthonormal splines with knots uniformly distributed according to the measure with density 1/ϕ. In case \gL{ina, p} is contained in the space of continuous functions, we give a discrete characterization of this space, using only values of function at the appropriate knots. Application of these results to characterize the order of polynomial approximation is presented.