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Construction of \(P^{th}\)-Stage Nonuniform Discrete Wavelet Frames

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Abstract

Motivated by the work of Frazier; and Gabardo and Nashed, we study \(P^{th}\)-stage nonuniform discrete wavelet frames (\(P^{th}\)-stage NUDW frames, in short) for \(\ell ^2(\Lambda )\), a nonuniform discrete space. In nonuniform discrete wavelet frames, the translation set is not necessary a group but a spectrum which is based on the theory of spectral pairs. We characterize first-stage nonuniform discrete Bessel sequences and wavelet frames in nonuniform discrete spaces. Duality and stability of first-stage NUDW frames are also discussed. Finally, by using first-stage NUDW frames, we provide a suitable way to construct the \(P^{th}\)-stage NUDW frames. We illustrate our construction with the help of a concrete example.

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Acknowledgements

The authors are grateful to two anonymous referees for carefully reading the manuscript, detecting many mistakes, and for offering valuable comments and suggestions which enabled the authors to substantially improve the paper.

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  1. Department of Mathematics, University of Delhi, Delhi, 110007, India

    Hari Krishan Malhotra & Lalit Kumar Vashisht

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  1. Hari Krishan Malhotra
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  2. Lalit Kumar Vashisht
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Correspondence to Lalit Kumar Vashisht.

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The research of first author is supported by the University Grants Commission (UGC), India (Grant No.: 19/06/2016(i)EU-V). The second author is supported by the Faculty Research Programme Grant-IoE, University of Delhi, Delhi-110007, India (Grant No.: IoE/FRP/PCMS/2020/27)

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Malhotra, H.K., Vashisht, L.K. Construction of \(P^{th}\)-Stage Nonuniform Discrete Wavelet Frames. Results Math 76, 117 (2021). https://doi.org/10.1007/s00025-021-01427-0

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