球面卷积算子逼近

研究d维欧氏空间R~d中单位球面上卷积算子的逼近问题.利用球面乘子理论以及K-泛函与光滑模等价关系,建立一类球面卷积算子逼近的正、逆定理.特别地,给出了逼近的强型逆向不等式,从而揭示了该类球面卷积算子的本质逼近阶.此外,作为应用,给出了球面Jackson-Matsuoka卷积算子与Abel-Poisson卷积算子逼近上、下界的相同阶估计.

Approximation by Spherical Convolution Operators

This paper studies the approximation of convolution operators defined on the unit sphere in d-dimensional Euclidean space R^d. By using the multiplier theory and the equivalence relation between K-functional and modulus of smoothness, the direct and converse theorems of the approximation by a class of spherical convolution operators are investigated. In particular, the inverse inequalities of the approximation of strong type are established, and thus the essential order of approximation for the convolution operators is reflected. As applications of the obtained main results, the estimates of the same orders of approximation bounds for the spherical Jackson–Matsuoka operators and spherical Abel–Poisson operators are given, respectively.