Kronecker product bases and their applications in approximation theory

Mohra Zayed; Gamal Hassan; Mohra Zayed; Gamal Hassan

doi:10.3934/era.2025048

Electronic Research Archive

2025, Volume 33, Issue 2: 1070-1092. doi: 10.3934/era.2025048

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Research article

Kronecker product bases and their applications in approximation theory

Mohra Zayed ^{1
,
,},
Gamal Hassan ²

1.
Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia
2.
Mathematics Department, Faculty of Science, University of Assiut, Assiut 71516, Egypt

Received: 13 November 2024 Revised: 24 January 2025 Accepted: 18 February 2025 Published: 26 February 2025

The Kronecker product is widely utilized to construct higher-dimensional spaces from lower-dimensional ones, making it an indispensable tool for efficiently analyzing multi-dimensional systems across various fields. This paper investigates the representation of analytic functions within hyper-elliptical regions through infinite series expansions involving sequences of Kronecker product bases of polynomials. Additionally, we examine the growth order and type and $ T_{\rho} $-property of series composed of Kronecker product bases that represent entire functions. We also delve into the convergence properties of Kronecker product bases associated with special functions, including Bessel, Chebyshev, Bernoulli, Euler, and Gontcharoff polynomials. The obtained results extend and enhance the existing findings of such representations in hyper-spherical regions.
- bases of polynomials,
- Kronecker product bases,
- growth of bases,
- effectiveness,
- hyper-elliptical regions
Citation: Mohra Zayed, Gamal Hassan. Kronecker product bases and their applications in approximation theory[J]. Electronic Research Archive, 2025, 33(2): 1070-1092. doi: 10.3934/era.2025048

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Abstract

The Kronecker product is widely utilized to construct higher-dimensional spaces from lower-dimensional ones, making it an indispensable tool for efficiently analyzing multi-dimensional systems across various fields. This paper investigates the representation of analytic functions within hyper-elliptical regions through infinite series expansions involving sequences of Kronecker product bases of polynomials. Additionally, we examine the growth order and type and $ T_{\rho} $-property of series composed of Kronecker product bases that represent entire functions. We also delve into the convergence properties of Kronecker product bases associated with special functions, including Bessel, Chebyshev, Bernoulli, Euler, and Gontcharoff polynomials. The obtained results extend and enhance the existing findings of such representations in hyper-spherical regions.

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