Abstract
Due to \(\mathbb R_+=(0,\,\infty )\) not being a group under addition, \(L^2(\mathbb R_+)\) admits no traditional wavelet or Gabor frames. This paper addresses a class of modulation-dilation frames (\(\mathcal MD\)-frames) for \(L^2(\mathbb R_+)\). We obtain a \(\Theta \)-transform matrix-based expression of adding generators to generate \(\mathcal MD\)-tight frames from a \(\mathcal{M}\mathcal{D}\)-Bessel sequences in \(L^2({\mathbb R}_+)\); and present criteria on \(\Phi \) with \(\mathcal{M}\mathcal{D}(\Psi \cup \Phi ,\,a,\,b)\) being a Parseval frame (an orthonormal basis) for an arbitrary Parseval frame sequence (an orthonormal sequence) \(\mathcal{M}\mathcal{D}(\Psi ,\,a,\,b)\) in \(L^2(\mathbb R_+)\). Some examples are also presented.
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The authors would like to thank the referees for reading this article carefully and providing valuable suggestions and comments which greatly improve the readability of this article. This work was supported by National Natural Science Foundation of China (Grant No. 11971043).
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This work was supported by National Natural Science Foundation of China (Grant No. 11971043).
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Li, YN., Li, YZ. Extensions of modulation-dilation Bessel Systems in \(L^2({\mathbb R}_+)\). Collect. Math. 75, 361–377 (2024). https://doi.org/10.1007/s13348-022-00389-y
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DOI: https://doi.org/10.1007/s13348-022-00389-y