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Windowed special affine Fourier transform

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Abstract

With the aim to circumvent the limitations of the special affine Fourier transform, we introduce a novel time–frequency transform namely the windowed special affine Fourier transform. We initiate our investigation by studying some fundamental properties of the proposed transform such as orthogonality relation, inversion formula and characterization of the range by employing the machinery of special affine Fourier transforms and operator theory. Continuing our endeavour, we propose a discrete analogue of the proposed windowed special affine Fourier transform and obtain the corresponding reconstruction formula. Besides, some potential applications of this new transform including windowed series expansion, Poisson summation formula, Paley–Wiener criterion and uncertainty principles are also given.

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Acknowledgements

The authors would like to extend sincere thanks to the anonymous referees’ for meticulously reading the manuscript and providing insightful revision suggestions thereon. The first author is supported by SERB (DST), Government of India, under Grant No. EMR/2016/007951.

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Authors and Affiliations

  1. Department of Mathematics, University of Kashmir, South Campus, Anantnag, Jammu and Kashmir, 192101, India

    Firdous A. Shah, Aajaz A. Teali & Azhar Y. Tantary

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  1. Firdous A. Shah
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  2. Aajaz A. Teali
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  3. Azhar Y. Tantary
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Correspondence to Firdous A. Shah.

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Shah, F.A., Teali, A.A. & Tantary, A.Y. Windowed special affine Fourier transform. J. Pseudo-Differ. Oper. Appl. 11, 1389–1420 (2020). https://doi.org/10.1007/s11868-019-00319-w

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