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How many Fourier samples are needed for real function reconstruction? (English) Zbl 1296.42001
Summary: In this paper we present some new results on the reconstruction of structured functions by a small number of equidistantly distributed Fourier samples. In particular, we show that real spline functions of order \(m\) with non-uniform knots containing \(N\) terms can be uniquely reconstructed by only \(m+N\) Fourier samples. Further, linear combinations of \(N\) non-equispaced shifts of a known low-pass function \(\varPhi\) can be reconstructed by \(N+1\) Fourier samples. In the bivariate case, we consider the problem of function recovering by a small amount of Fourier samples on different lines through the origin. Our methods are based on the Prony method. The proofs given in this paper are constructive. Some numerical examples show the applicability of the proposed approach.

42A10 Trigonometric approximation
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
65T40 Numerical methods for trigonometric approximation and interpolation
41A45 Approximation by arbitrary linear expressions
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
Full Text: DOI
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