Remarks on the $L^p$ convergence of Bessel--Fourier series on the disc
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916104033533952 |
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| author | Babb, Ryan L. Acosta |
| author_facet | Babb, Ryan L. Acosta |
| contents | The $L^p$ convergence of eigenfunction expansions for the Laplacian on planar domains is largely unknown for $p\neq 2$. After discussing the classical Fourier series on the 2-torus, we move onto the disc, whose eigenfunctions are explicitly computable as products of trigonometric and Bessel functions. We summarise a result of Balodis and Córdoba (1999) regarding the $L^p$ convergence of the Bessel--Fourier series in the mixed norm space $L^p_{\mathrm{rad}}(L^2_{\mathrm{ang}})$ on the disk for the range $\tfrac{4}{3}<p<4$. We then describe how to modify their result to obtain $L^p(\mathbb{D}, r\,\mathrm{d}r\,\mathrm{d}t)$ norm convergence in the subspace $L^p_{\mathrm{rad}}(L^q_{\mathrm{ang}})$($\tfrac{1}{p}+\tfrac{1}{q}=1$) for the restricted range ${2\leq p < 4}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_06119 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Remarks on the $L^p$ convergence of Bessel--Fourier series on the disc Babb, Ryan L. Acosta Classical Analysis and ODEs Analysis of PDEs Functional Analysis 42-02 (Primary) 42C05, 33C10, 34L10, (Secondary) The $L^p$ convergence of eigenfunction expansions for the Laplacian on planar domains is largely unknown for $p\neq 2$. After discussing the classical Fourier series on the 2-torus, we move onto the disc, whose eigenfunctions are explicitly computable as products of trigonometric and Bessel functions. We summarise a result of Balodis and Córdoba (1999) regarding the $L^p$ convergence of the Bessel--Fourier series in the mixed norm space $L^p_{\mathrm{rad}}(L^2_{\mathrm{ang}})$ on the disk for the range $\tfrac{4}{3}<p<4$. We then describe how to modify their result to obtain $L^p(\mathbb{D}, r\,\mathrm{d}r\,\mathrm{d}t)$ norm convergence in the subspace $L^p_{\mathrm{rad}}(L^q_{\mathrm{ang}})$($\tfrac{1}{p}+\tfrac{1}{q}=1$) for the restricted range ${2\leq p < 4}$. |
| title | Remarks on the $L^p$ convergence of Bessel--Fourier series on the disc |
| topic | Classical Analysis and ODEs Analysis of PDEs Functional Analysis 42-02 (Primary) 42C05, 33C10, 34L10, (Secondary) |
| url | https://arxiv.org/abs/2202.06119 |