Bibliographic Information: Jyoti and Lalit Kumar Vashisht. (2024). Operators for matrix-valued Riesz bases over LCA groups. arXiv preprint arXiv:2410.09446v1.
Research Objective: This paper investigates the properties and construction methods of matrix-valued Riesz bases within the function space L2(G, Cs×r) over locally compact abelian groups (LCA). The authors aim to identify classes of operators that can generate these bases from existing orthonormal bases.
Methodology: The authors utilize functional analysis techniques, particularly operator theory, to analyze the properties of operators acting on L2(G, Cs×r). They leverage concepts like positive operators, self-adjoint operators, and the polar decomposition of operators to establish conditions for generating Riesz bases.
Key Findings: The research establishes that specific classes of operators, including positive operators, powers of positive operators, and operators related to self-adjoint operators, can be employed to construct matrix-valued Riesz bases. Notably, the paper demonstrates that the image of a matrix-valued orthonormal basis under a bounded, linear, and bijective operator on L2(G, Cs×r) might not always form a frame, highlighting a key difference from standard Riesz bases in separable Hilbert spaces.
Main Conclusions: The study successfully identifies several classes of operators capable of generating matrix-valued Riesz bases in L2(G, Cs×r). It emphasizes the significance of operator properties like positivity and self-adjointness in this construction process. Additionally, the research provides insights into the relationship between matrix-valued Riesz bases and frames in this function space.
Significance: This research contributes significantly to the understanding of matrix-valued Riesz bases, which are fundamental in areas like signal processing and harmonic analysis. By characterizing operators that generate these bases, the study offers valuable tools for constructing and analyzing signals and functions in L2(G, Cs×r) spaces.
Limitations and Future Research: The paper primarily focuses on theoretical aspects of matrix-valued Riesz bases. Further research could explore practical applications of these findings, particularly in signal processing, wavelet analysis, and other areas where matrix-valued functions over LCA groups are relevant. Investigating the numerical stability and computational efficiency of the proposed construction methods would be beneficial for real-world implementations.
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by Jyoti, Lalit... at arxiv.org 10-15-2024
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