A Battle-Lemarié Frame Characterization of Besov and Triebel-Lizorkin Spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915890070552576 |
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| author | Haar, Andrew |
| author_facet | Haar, Andrew |
| contents | In this paper, we investigate a spline frame generated by oversampling against the well-known Battle-Lemarié wavelet system of nonnegative integer order, $n$. We establish a characterization of the Besov and Triebel-Lizorkin (quasi-) norms for the smoothness parameter up to $s < n+1$, which includes values of $s$ where the Battle-Lemarié system no longer provides an unconditional basis; we, additionally, prove a result for the endpoint case $s=n+1$. This builds off of earlier work by G. Garrigós, A. Seeger, and T. Ullrich, where they proved the case $n=0$, i.e. that of the Haar wavelet, and work of R. Srivastava, where she gave a necessary range for the Battle-Lemarié system to give an unconditional basis of the Triebel-Lizorkin spaces. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_12783 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Battle-Lemarié Frame Characterization of Besov and Triebel-Lizorkin Spaces Haar, Andrew Functional Analysis Classical Analysis and ODEs In this paper, we investigate a spline frame generated by oversampling against the well-known Battle-Lemarié wavelet system of nonnegative integer order, $n$. We establish a characterization of the Besov and Triebel-Lizorkin (quasi-) norms for the smoothness parameter up to $s < n+1$, which includes values of $s$ where the Battle-Lemarié system no longer provides an unconditional basis; we, additionally, prove a result for the endpoint case $s=n+1$. This builds off of earlier work by G. Garrigós, A. Seeger, and T. Ullrich, where they proved the case $n=0$, i.e. that of the Haar wavelet, and work of R. Srivastava, where she gave a necessary range for the Battle-Lemarié system to give an unconditional basis of the Triebel-Lizorkin spaces. |
| title | A Battle-Lemarié Frame Characterization of Besov and Triebel-Lizorkin Spaces |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2504.12783 |