On the concentration of the Fourier coefficients for products of Laplace-Beltrami eigenfunctions on real-analytic manifolds
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| Format: | Preprint |
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2024
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| _version_ | 1866929292065112064 |
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| author | Charron, Philippe Pagano, François |
| author_facet | Charron, Philippe Pagano, François |
| contents | On a closed analytic manifold $(M,g)$, let $ϕ_i$ be the eigenfunctions of $Δ_g$ with eigenvalues $λ_i^2$ and let $f:=\prod ϕ_{k_j}$ be a finite product of Laplace-Beltrami eigenfunctions. We show that $\left\langle f, ϕ_i \right\rangle_{L^2(M)}$ decays exponentially as soon as $λ_i > C \sum λ_{k_j}$ for some constant $C$ depending only on $M$. Moreover, by using a lower bound on $\| f \|_{L^2(M)} $, we show that $99\%$ of the $L^2$-mass of $f$ can be recovered using only finitely many Fourier coefficients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_18663 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the concentration of the Fourier coefficients for products of Laplace-Beltrami eigenfunctions on real-analytic manifolds Charron, Philippe Pagano, François Analysis of PDEs Classical Analysis and ODEs Spectral Theory 35C99, 35A10, 35P10 On a closed analytic manifold $(M,g)$, let $ϕ_i$ be the eigenfunctions of $Δ_g$ with eigenvalues $λ_i^2$ and let $f:=\prod ϕ_{k_j}$ be a finite product of Laplace-Beltrami eigenfunctions. We show that $\left\langle f, ϕ_i \right\rangle_{L^2(M)}$ decays exponentially as soon as $λ_i > C \sum λ_{k_j}$ for some constant $C$ depending only on $M$. Moreover, by using a lower bound on $\| f \|_{L^2(M)} $, we show that $99\%$ of the $L^2$-mass of $f$ can be recovered using only finitely many Fourier coefficients. |
| title | On the concentration of the Fourier coefficients for products of Laplace-Beltrami eigenfunctions on real-analytic manifolds |
| topic | Analysis of PDEs Classical Analysis and ODEs Spectral Theory 35C99, 35A10, 35P10 |
| url | https://arxiv.org/abs/2403.18663 |