Constructing Multiresolution Analysis via Wavelet Packets on Sobolev Space in Local Fields
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910549066907648 |
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| author | Kumar, Manish |
| author_facet | Kumar, Manish |
| contents | We define Sobolev spaces $H^{\mathfrak{s}}(K_q)$ over a local field $K_q$ of finite characteristic $p>0$, where $q=p^c$ for a prime $p$ and $c\in \mathbb{N}$. This paper introduces novel fractal functions, such as the Weierstrass type and 3-adic Cantor type, as intriguing examples within these spaces and a few others. Employing prime elements, we develop a Multi-Resolution Analysis (MRA) and examine wavelet expansions, focusing on the orthogonality of both basic and fractal wavelet packets at various scales. We utilize convolution theory to construct Haar wavelet packets and demonstrate the orthogonality of all discussed wavelet packets within $H^{\mathfrak{s}}(K_q)$, enhancing the analytical capabilities of these Sobolev spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00028 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructing Multiresolution Analysis via Wavelet Packets on Sobolev Space in Local Fields Kumar, Manish Rings and Algebras 42C40, 40A30, 11F87, 43A70, 46E35 We define Sobolev spaces $H^{\mathfrak{s}}(K_q)$ over a local field $K_q$ of finite characteristic $p>0$, where $q=p^c$ for a prime $p$ and $c\in \mathbb{N}$. This paper introduces novel fractal functions, such as the Weierstrass type and 3-adic Cantor type, as intriguing examples within these spaces and a few others. Employing prime elements, we develop a Multi-Resolution Analysis (MRA) and examine wavelet expansions, focusing on the orthogonality of both basic and fractal wavelet packets at various scales. We utilize convolution theory to construct Haar wavelet packets and demonstrate the orthogonality of all discussed wavelet packets within $H^{\mathfrak{s}}(K_q)$, enhancing the analytical capabilities of these Sobolev spaces. |
| title | Constructing Multiresolution Analysis via Wavelet Packets on Sobolev Space in Local Fields |
| topic | Rings and Algebras 42C40, 40A30, 11F87, 43A70, 46E35 |
| url | https://arxiv.org/abs/2408.00028 |