A priori \(L^2\)-error estimates for approximations of functions on compact manifolds. (English) Zbl 1315.58007
Given a \(C^2\)-function \(f\) on a compact Riemannian manifold \((X,g),\) the authors give a set of frequencies \(L=L_f(\varepsilon)\) depending on a small parameter \(\varepsilon>0\) such that the relative \(L^2\)-error \(\frac{\|f-f^L\|}{\|f\|}\) is bounded from above by \(\varepsilon,\) where \(f^L\) denotes the \(L\)-partial sum of the Fourier series \(f\) with respect to an orthonormal basis of \(L^2 (X)\) constituted by eigenfunctions of the Laplace-Beltrami operator \(\Delta_g\) associated to the metric \(g\).
Reviewer: Dian K. Palagachev (Bari)
MSC:
| 58C40 | Spectral theory; eigenvalue problems on manifolds |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
Keywords:
Fourier analysis; Riemannian manifolds; Laplacian operator; spherical harmonics; approximation theoryReferences:
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