Abstract
In this paper, we study the pointwise convergence of the Calderón reproducing formula, which is also known as an inversion formula for wavelet transforms. We show that for every \(f\in L_{w}^{p}(\mathbb {R}^{d})\) with an \(\mathcal{A}_{p}\) weight w, 1≤p<∞, the integral is convergent at every Lebesgue point of f, and therefore almost everywhere. Moreover, we prove the convergence without any assumption on the smoothness of wavelet functions.
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The authors thank the referees and Professor Hans G. Feichtinger for elaborate and valuable suggestions which helped to improve this paper.
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Communicated by Hans G. Feichtinger.
This work was supported partially by the National Natural Science Foundation of China (10971105 and 10990012) and the Natural Science Foundation of Tianjin (09JCYBJC01000).
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Li, K., Sun, W. Pointwise Convergence of the Calderón Reproducing Formula. J Fourier Anal Appl 18, 439–455 (2012). https://doi.org/10.1007/s00041-011-9211-4
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DOI: https://doi.org/10.1007/s00041-011-9211-4