Equivalence between K-functionals and modulus of smoothness on the quaternion algebra

摘要

Abstract In the space L2(R2,H)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$L^{2}({mathbb {R}}^{2},{mathcal {H}})$$end{document}, using the analog of the Steklov operator, we construct the generalized modulus of smoothness, and also using the Laplacian operator we define the K-functional. The main result of our article is the proof of the equivalence between K-functionals and modulus of smoothness.