Moduli of Smoothness and Approximation on the Unit Sphere and the Unit Ball

TL;DR

This paper introduces a new modulus of smoothness based on Euler angles for the unit sphere, establishing its properties and extending it to the unit ball to characterize polynomial approximation.

Contribution

It presents a novel modulus of smoothness on the sphere, proves its key properties, and extends the concept to the ball for approximation analysis.

Findings

The new modulus satisfies characteristic properties of smoothness measures.

Establishes direct and inverse theorems for polynomial approximation.

Defines equivalent $K$-functionals for smoothness and approximation.

Abstract

A new modulus of smoothness based on the Euler angles is introduced on the unit sphere and is shown to satisfy all the usual characteristic properties of moduli of smoothness, including direct and inverse theorem for the best approximation by polynomials and its equivalence to a $K$-functional, defined via partial derivatives in Euler angles. The set of results on the moduli on the sphere serves as a basis for defining new moduli of smoothness and their corresponding $K$-functionals on the unit ball, which are used to characterize the best approximation by polynomials on the ball.