空间 $A^2$ 和 $\mathcal{D}^2$ 上具有 Hyers-Ulam 稳定性的系数乘子

数学物理学报 ›› 2025, Vol. 45 ›› Issue (5): 1381-1391.

空间 $A^2$ 和 $\mathcal{D}^2$ 上具有 Hyers-Ulam 稳定性的系数乘子

王春^1,^*(),许天周²

¹长治学院数学系山西长治 046011
²北京理工大学数学与统计学院北京 100081

收稿日期:2024-02-23 修回日期:2025-05-05 出版日期:2025-10-26 发布日期:2025-10-14
通讯作者: * 王春,E-mail:wangchun12001@163.com
基金资助:
山西省基础研究计划(202203021211110)

The Coefficient Multipliers Between $A^2$ and $\mathcal{D}^2$ with Hyers-Ulam Stability

Chun Wang^1,^*(),Tianzhou Xu²

¹Department of Mathematics, Changzhi University, Shanxi Changzhi 046011
²School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081

Received:2024-02-23 Revised:2025-05-05 Online:2025-10-26 Published:2025-10-14
Supported by:
Fundamental Research Program of Shanxi Province(202203021211110)

摘要/Abstract

摘要：

该文研究了 Bergman 空间和 Dirichlet 空间上系数乘子的 Hyers-Ulam 稳定性. 对于一个复序列 $\lambda=\{\lambda_n\}_{n=0}^\infty$ 能否作为空间 $A^2$ 和 $\mathcal{D}^2$ 之间的系数乘子, 给出了一些充分条件. 该文给出了一些在 Bergman 空间 $A^2$ 和 Dirichlet 空间 $\mathcal{D}^2$ 之间的系数乘子具有 Hyers-Ulam 稳定性的充分必要条件. 同时, 证明了在不同情形下的最佳 Hyers-Ulam 稳定性常数是存在的. 此外, 讨论了一些与主要结果相关的例子.

关键词: 系数乘子, Hyers-Ulam 稳定性, Dirichlet 空间, Bergman 空间

Abstract:

In this paper, Hyers-Ulam stability of the coefficient multipliers between Bergman space $A^2$ and Dirichlet space $\mathcal{D}^2$ are investigated. Some sufficient conditions which for that a complex sequence $\lambda=\{\lambda_n\}_{n=0}^\infty$ can be the coefficient multiplier between Bergman space $A^2$ and Dirichlet space $\mathcal{D}^2$ are given. Some necessary and sufficient conditions for that the coefficient multipliers have the Hyers-Ulam stability between Bergman space $A^2$ and Dirichlet space $\mathcal{D}^2$ are also given. This paper also show that the best constant of Hyers-Ulam stability exists under different circumstances. Moreover, some illustrative examples are also discussed.

Key words: coefficient multipliers, Hyers-Ulam stability, Dirichlet spaces, Bergman spaces

中图分类号:

O177.9

王春, 许天周. 空间 $A^2$ 和 $\mathcal{D}^2$ 上具有 Hyers-Ulam 稳定性的系数乘子[J]. 数学物理学报, 2025, 45(5): 1381-1391.

Chun Wang, Tianzhou Xu. The Coefficient Multipliers Between $A^2$ and $\mathcal{D}^2$ with Hyers-Ulam Stability[J]. Acta mathematica scientia,Series A, 2025, 45(5): 1381-1391.

参考文献

[1]	Blasco O. Multipliers on spaces of analytic functions. Canad J Math, 1995, 47: 44-64
[2]	Brzd?k J, Jung S M. A note on stability of an operator linear equation of the second order. Abstr Appl Anal, 2011, 2011: Article 602713
[3]	Brzd?k J, Popa D, Xu B. On approximate solutions of the linear functional equation of higher order. J Math Anal Appl, 2011, 373: 680-689
[4]	Ciepliński K. Applications of fixed point theorems to the Hyers-Ulam stability of functional equations - a survey. Ann Funct Anal, 2012, 3: 151-164
[5]	Ciepliński K. Generalized stability of multi-additive mappings. Appl Math Lett, 2010, 23: 1291-1294
[6]	Duren P L. On the multipliers of $H^p$ spaces. Proc Amer Math Soc, 1969, 22: 24-27
[7]	Duren P L. Theory of H^p Spaces. New York: Academic Press, 1970
[8]	G?vru?a P. A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl, 1994, 184: 431-436
[9]	Hardy G H, Littlewood J E. Some properties of fractional integrals Ⅱ. Math Z, 1932, 34: 403-439
[10]	Hatori O, Kobayasi K, Miura T, et al. On the best constant of Hyers-Ulam stability. J Nonlinear Convex Anal, 2004, 5: 387-393
[11]	Hirasawa G, Miura T. Hyers-Ulam stability of a closed operator in a Hilbert space. Bull Korean Math Soc, 2006, 43: 107-117
[12]	Hyers D H. On the stability of the linear functional equation. Proc Natl Acad Sci USA, 1941, 27: 222-224
[13]	Macgregor T, Zhu K. Coefficient multipliers between Bergman and Hardy spaces. Mathematika, 1995, 42: 413-426
[14]	Mateljevic M, Pavlovic M. Multipliers of $H^p$ and BMOA. Pacific J Math, 1990, 146: 71-84
[15]	Miura T, Hirasawa G, Takahasi S E. Ger-type and Hyers-Ulam stabilities for the first-order linear differential opetators of entire functions. Int J Math Math Sci, 2004, 2004: 1151-1158
[16]	Miura T, Hirasawa G, Takahasi S E. Stability of multipliers on Banach algebras. Int J Math Math Sci, 2004, 2004: 2377-2381
[17]	Popa D, Ra?a I. The Fréchet functional equation with application to the stability of certain operators. J Approx Theory, 2012, 164: 138-144
[18]	Popa D, Ra?a I. On the stability of some classical operators from approximation theory. Expo Math, 2013, 31: 205-214
[19]	Rassias Th M. On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc, 1978, 72: 297-300
[20]	Sledd W T. On multipliers of $H^p$ spaces. Indiana Univ Math J, 1978, 27: 797-803
[21]	Takagi H, Miura T, Takahasi S E. Essential norms and stability constants of weighted composition operators on $C(X)$. Bull Korean Math Soc, 2003, 40: 583-591
[22]	Ulam S M. A Collection of Mathematical Problems. Tracts in Pure and Apllied Mathematics, Number 8. New York:Interscience Publishers, 1960
[23]	Vukotic D. On the coefficient multipliers of Bergman spaces. J Lond Math Soc, 1994, 50: 341-348
[24]	Wang C, Xu T Z. Hyers-Ulam stability of differentiation operator on Hilbert spaces of entire functions. J Funct Spaces, 2014, 2014: Article 398673
[25]	Wang C, Xu T Z. Hyers-Ulam stability of differential operators on reproducing kernel function spaces. Complex Anal Oper Theory, 2016, 10: 795-813
[26]	王春, 许天周. 2-Banach 空间上三次-四次混合型函数方程的Hyers-Vtam-Rassias 稳定性. 数学物理学报, 2020, 40A(2): 352-368
[26]	Wang C, Xu T Z. Hyers-Vtam-Rassias stability of a mixed type cubic-quartic functional equation in 2-Banach spaces. Acta Math Sci, 2020, 40A(2): 352-368
[27]	Wojtaszczyk P. On multipliers into Bergman spaces and Nevanlinna classes. Canad Math Bull, 1990, 33: 151-161
[28]	Wu Z, Yang L. Multipliers between Dirichlet spaces. Integral Equations Operator Theory, 1998, 32: 482-492
[29]	Xu T Z. On the stability of multi-Jensen mappings in $\beta$-normed spaces. Appl Math Lett, 2012, 25: 1866-1870
[30]	Xu T Z. On fuzzy approximately cubic type mapping in fuzzy Banach spaces. Inform Sci, 2014, 278: 56-66
[31]	Xu T Z, Wang C, Rassias Th M. On the stability of multi-additive mappings in non-Archimedean normed spaces. J Comput Anal Appl, 2015, 18: 1102-1110
[32]	Xu T Z, Yang Z, Rassias J M. Direct and fixed point approaches to the stability of an AQ-functional equation in non-Archimedean normed spaces. J Comput Anal Appl, 2014, 17: 697-706