A distributional framework for the short-time special affine Fourier transform and quasiasymptotic analysis

Astrit Ferizi; Katerina Hadzi-Velkova Saneva; Astrit Ferizi; Katerina Hadzi-Velkova Saneva

doi:10.3934/math.20261333

AIMS Mathematics

2026, Volume 11, Issue 10: 34172-34192. doi: 10.3934/math.20261333

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Research article

A distributional framework for the short-time special affine Fourier transform and quasiasymptotic analysis

Astrit Ferizi ^{1
,
,},
Katerina Hadzi-Velkova Saneva ²

1.
Faculty of Mathematics and Natural Sciences, University of Prishtina, George Bush 31, Prishtina, 10000, Kosovo
2.
Faculty of Electrical Engineering and Information Technologies, Ss. Cyril and Methodius University, Rugjer Boshkovikj 18, Skopje, 1000, North Macedonia

Received: 20 July 2026 Revised: 31 August 2026 Accepted: 17 September 2026 Published: 20 October 2026
MSC : 26A12, 40E05, 42A38, 46F05, 46F12

Existing theory of the short-time special affine Fourier transform (STSAFT) is primarily formulated in the classical $ L^2 $-setting and does not provide a framework for tempered distributions and their quasiasymptotic analysis. To address this gap, we develop a distributional framework for the STSAFT. We establish the continuity of the STSAFT and its synthesis operator on the corresponding Schwartz spaces of test functions and extend both operators to the associated spaces of tempered distributions by duality. Within this framework, we derive a generalized inversion formula, a desingularization formula, and a characterization of bounded subsets of $ \mathcal{S}'(\mathbb{R}) $ in terms of polynomial growth estimates of the STSAFT. Furthermore, we establish several Abelian- and Tauberian-type results that relate the quasiasymptotic behavior and quasiasymptotic boundedness of tempered distributions at the origin to the scaled asymptotic behavior of their STSAFT. These results provide a rigorous basis for studying singular signals and local asymptotic phenomena, with potential relevance to distribution theory, time-frequency analysis, optics, and communications.
Citation: Astrit Ferizi, Katerina Hadzi-Velkova Saneva. A distributional framework for the short-time special affine Fourier transform and quasiasymptotic analysis[J]. AIMS Mathematics, 2026, 11(10): 34172-34192. doi: 10.3934/math.20261333

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Abstract

Existing theory of the short-time special affine Fourier transform (STSAFT) is primarily formulated in the classical $ L^2 $-setting and does not provide a framework for tempered distributions and their quasiasymptotic analysis. To address this gap, we develop a distributional framework for the STSAFT. We establish the continuity of the STSAFT and its synthesis operator on the corresponding Schwartz spaces of test functions and extend both operators to the associated spaces of tempered distributions by duality. Within this framework, we derive a generalized inversion formula, a desingularization formula, and a characterization of bounded subsets of $ \mathcal{S}'(\mathbb{R}) $ in terms of polynomial growth estimates of the STSAFT. Furthermore, we establish several Abelian- and Tauberian-type results that relate the quasiasymptotic behavior and quasiasymptotic boundedness of tempered distributions at the origin to the scaled asymptotic behavior of their STSAFT. These results provide a rigorous basis for studying singular signals and local asymptotic phenomena, with potential relevance to distribution theory, time-frequency analysis, optics, and communications.

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