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Operating Functions on \(A^q_s({{\mathbf {T}}})\)

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Abstract

Let \(A^q_s({{\mathbf {T}}})\) denote the space of all Lebesgue integrable functions f on the torus \({\mathbf {T}}\) such that \(\sum _{m \in {{\mathbf {Z}}}} |\widehat{f}(m)|^q \langle m \rangle ^{sq} < \infty \), where \(\{ \widehat{f}(m) \}_{m \in {{\mathbf {Z}}}}\) denote the Fourier coefficients of f. We consider necessary and sufficient conditions for all functions \(F \in A^1_\beta ({{\mathbf {T}}})\) to operate on all real-valued functions in \(A^q_s({{\mathbf {T}}})\).

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Acknowledgements

We would like to thank Professor Hans Georg Feichtinger for several valuable comments and suggestions which improve our paper. The authors also wish to thank the anonymous referees for providing some helpful comments.

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Correspondence to Masaharu Kobayashi.

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Communicated by Ferenc Weisz.

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This work was supported by JSPS KAKENHI Grant Numbers JP19K03533, JP17K05289.

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Kobayashi, M., Sato, E. Operating Functions on \(A^q_s({{\mathbf {T}}})\). J Fourier Anal Appl 28, 42 (2022). https://doi.org/10.1007/s00041-022-09925-7

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  • DOI: https://doi.org/10.1007/s00041-022-09925-7

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