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This chapter investigates various aspects of signals in mixed Lebesgue spaces \(L^{p,q}(\mathbb {R}^{d+1})\) . Firstly, the definition of mixed Lebesgue spaces is given, then the structures of shift-invariant spaces in \(L^{p,q}(\mathbb {R}^{d+1})\) are characterized in terms of semi-discrete convolutions of their generators with sequences in \(\ell ^{p,q}(\mathbb {Z}^{d+1})\) , and it is shown that if the generators are compactly supported, these spaces are the intersection of \(L^{p,q}(\mathbb {R}^{d+1})\) and the linear space of functions formed with these convolutions.

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Samplings in Mixed Lebesgue Spaces

  • 章节

  • 发表日期：

    2026年

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  • Deguang Han,
  • Qianfeng Hu,
  • Bei Liu,
  • Rui Liu

摘要

This chapter investigates various aspects of signals in mixed Lebesgue spaces \(L^{p,q}(\mathbb {R}^{d+1})\) . Firstly, the definition of mixed Lebesgue spaces is given, then the structures of shift-invariant spaces in \(L^{p,q}(\mathbb {R}^{d+1})\) are characterized in terms of semi-discrete convolutions of their generators with sequences in \(\ell ^{p,q}(\mathbb {Z}^{d+1})\) , and it is shown that if the generators are compactly supported, these spaces are the intersection of \(L^{p,q}(\mathbb {R}^{d+1})\) and the linear space of functions formed with these convolutions.