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Random sampling of sparse trigonometric polynomials. (English) Zbl 1123.94004
Summary: We study the problem of reconstructing a multivariate trigonometric polynomial having only few non-zero coefficients from few random samples. Inspired by recent work of E. J. Candès, J. K. Romberg and T. Tao [Commun. Pure Appl. Math. 59, No. 8, 1207–1223 (2006; Zbl 1098.94009)] we propose to recover the polynomial by Basis Pursuit, i.e., by \(\ell ^{1}\)-minimization. In contrast to their work, where the sampling points are restricted to a grid, we model the random sampling points by a continuous uniform distribution on the cube, i.e., we allow them to have arbitrary position. Numerical experiments show that with high probability the trigonometric polynomial can be recovered exactly provided the number \(N\) of samples is high enough compared to the “sparsity” – the number of non-vanishing coefficients. However, \(N\) can be chosen small compared to the assumed maximal degree of the trigonometric polynomial. We present two theorems that explain this observation. One of them provides the analogue of the result of Candès, Romberg and Tao. The other one is a result toward an average case analysis and, unexpectedly connects to an interesting combinatorial problem concerning set partitions, which seemingly has not yet been considered before. Although our proofs follow ideas of Candès et al. they are simpler.

MSC:
94A20 Sampling theory in information and communication theory
42A05 Trigonometric polynomials, inequalities, extremal problems
15B52 Random matrices (algebraic aspects)
Software:
OEIS; PDCO
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References:
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