Reconstructing Multivariate Trigonometric Polynomials by Sampling along Generated Sets
نویسنده
چکیده
The approximation of problems in d spatial dimensions by sparse trigonometric polynomials supported on known or unknown frequency index sets I ⊂ Zd is an important task with a variety of applications. The use of a generalization of rank1 lattices as spatial discretizations offers a suitable possibility for sampling such sparse trigonometric polynomials. Given an index set of frequencies, we construct corresponding sampling sets that allow a stable and unique discrete Fourier transform. Applying the one-dimensional non-equispaced fast Fourier transform (NFFT) enables the fast evaluation and reconstruction of the multivariate trigonometric polynomials.
منابع مشابه
Reconstructing hyperbolic cross trigonometric polynomials by sampling along generated sets
The evaluation of multivariate trigonometric polynomials at the nodes of a rank-1 lattice leads to a onedimensional discrete Fourier transform. Often, one is also interested in the reconstruction of the Fourier coefficients from their samples. We present necessary and sufficient conditions on rank-1 lattices allowing a stable reconstruction of trigonometric polynomials supported on hyperbolic c...
متن کاملReconstructing Hyperbolic Cross Trigonometric Polynomials by Sampling along Rank-1 Lattices
The evaluation of multivariate trigonometric polynomials at the nodes of a rank-1 lattice leads to a onedimensional discrete Fourier transform. Often, one is also interested in the reconstruction of the Fourier coefficients from their samples. We present necessary and sufficient conditions on rank-1 lattices allowing a stable reconstruction of trigonometric polynomials supported on hyperbolic c...
متن کاملMultiple Rank-1 Lattices as Sampling Schemes for Multivariate Trigonometric Polynomials
We present a new sampling method that allows the unique reconstruction of (sparse) multivariate trigonometric polynomials. The crucial idea is to use several rank-1 lattices as spatial discretization in order to overcome limitations of a single rank-1 lattice sampling method. The structure of the corresponding sampling scheme allows for the fast computation of the evaluation and the reconstruct...
متن کاملApproximation of multivariate periodic functions by trigonometric polynomials based on sampling along rank-1 lattice with generating vector of Korobov form
In this paper, we present error estimates for the approximation of multivariate periodic functions in periodic Hilbert spaces of isotropic and dominating mixed smoothness by trigonometric polynomials. The approximation is based on sampling of the multivariate functions on rank-1 lattices. We use reconstructing rank-1 lattices with generating vectors of Korobov form for the sampling and generali...
متن کاملRandom Sampling of Sparse Trigonometric Polynomials, II. Orthogonal Matching Pursuit versus Basis Pursuit
We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction via Thresholding and ...
متن کامل