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On the concentration of the Fourier coefficients for products of Laplace-Beltrami eigenfunctions on real-analytic manifolds

Philippe Charron, François Pagano

Abstract

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.

On the concentration of the Fourier coefficients for products of Laplace-Beltrami eigenfunctions on real-analytic manifolds

Abstract

On a closed analytic manifold , let be the eigenfunctions of with eigenvalues and let be a finite product of Laplace-Beltrami eigenfunctions. We show that decays exponentially as soon as for some constant depending only on . Moreover, by using a lower bound on , we show that of the -mass of can be recovered using only finitely many Fourier coefficients.
Paper Structure (24 sections, 17 theorems, 75 equations)

This paper contains 24 sections, 17 theorems, 75 equations.

Table of Contents

  1. Introduction and main results
  2. Main results
  3. Acknowledgements
  4. Definitions and notation
  5. The manifold:
  6. Multi-indices
  7. Partial derivatives
  8. Polydisks
  9. Power series
  10. Laplace-Beltrami operator
  11. Proof of Theorem \ref{['maintheorem']}
  12. Outline of the proof
  13. Radius of holomorphic extension of the coefficients of the Laplacian
  14. Holomorphic extension of Laplace eigenfunction
  15. Harmonic extension of the product of the complexified eigenfunctions
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(M, g)$ be a real-analytic closed Riemannian manifold of dimension $d$. Let $\{\phi_k\}$ be an orthonormal basis of Laplace-Beltrami eigenfunctions on $M$ with eigenvalues $\lambda_k^2$ in increasing order. Let $n \geq 2$ and consider a finite set of positive indices $\{{k_1}, \dots, {k_n}\}$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Remark 4.3
  • ...and 18 more