球面卷积算子逼近

丁春梅, 曹飞龙

数学学报 ›› 2015, Vol. 58 ›› Issue (6) : 1009-1020.

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数学学报 ›› 2015, Vol. 58 ›› Issue (6) : 1009-1020. DOI: 10.12386/A2015sxxb0100
论文

球面卷积算子逼近

    丁春梅, 曹飞龙
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Approximation by Spherical Convolution Operators

    Chun Mei DING, Fei Long CAO
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文章历史 +

摘要

研究d维欧氏空间Rd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.

Abstract

This paper studies the approximation of convolution operators defined on the unit sphere in d-dimensional Euclidean space Rd. 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.

关键词

球面 / 卷积 / 逼近 / 正定理 / 逆定理

Key words

sphere / convolution / approximation / direct theorem / inverse theorem

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丁春梅, 曹飞龙. 球面卷积算子逼近. 数学学报, 2015, 58(6): 1009-1020 https://doi.org/10.12386/A2015sxxb0100
Chun Mei DING, Fei Long CAO. Approximation by Spherical Convolution Operators. Acta Mathematica Sinica, Chinese Series, 2015, 58(6): 1009-1020 https://doi.org/10.12386/A2015sxxb0100

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基金

国家自然科学基金资助项目(61272023)

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