Titchmarsh Theorems and K-Functionals for the Two-Sided Quaternion Fourier Transform

The purpose of this paper is to study the Quaternion Fourier transforms of functions that satisfy Lipschitz conditions of certain orders. Thus we study the Quaternion Fourier transforms of Lipschitz function in the functions space Lr(R2; H), where H a quaternion algebra which will be specified in due course. Our investigation into the problem was motivated by a theorem proved by Titchmarsh [[29], Theorem 85] for Lipschitz functions on the real line. we will give also some results on calculation of the K-functional which have number of applications of interpolation theory. In particular some recent problems in image processing and singular integral operators require the computation of suitable K-functionals. In this paper we will give some results concerning the equivalence of a K-functional and the modulus of smoothness constructed by the Steklov function