Equivalence of K-Functionals and Modulus of Smoothness Constructed by the Mehler-Fock-Clifford Transform

Abstract

Using the Mehelt-Fock-Clifford transform, we define generalized modulus of smoothness in the space \(L^2(J;x^{-\frac{1}{2}}dx)\). Based on the kernel \(P_{i\sqrt{\lambda}-\frac{1}{2}}\) and the operator \(A_x^m\) we define Sobolev-type and \(K\)-functionals. The main result of the paper is the proof of the equivalence theorem for a \(K\)-functional and a modulus of smoothness for the Mehler-Fock-Clifford transform.