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Reconstruction of spline spectra-signals from generalized sinc function by finitely many samples. (English) Zbl 1457.42054

Summary: Reconstruction of signals by their Fourier (transform) samples is investigated in many mathematical/engineering problems such as the inverse Radon transform and optical diffraction tomography. This paper concerns on the reconstruction of spline-spectra signals in \(V(\mathrm{sinc}_a)\) by finitely many Fourier samples, where \(\mathrm{sinc}_a\) is the generalized sinc function. There are two main results on this topic. When the spectra knots are known, the exact reconstruction formula conducted by finitely many Fourier samples is established in the first main theorem. When the spectra knots are unknown, in the second main theorem we establish the approximations to the spline-spectra signals also by finitely many Fourier samples. Numerical simulations are conducted to check the efficiency of the approximation.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A30 Approximation by other special function classes
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
94A20 Sampling theory in information and communication theory
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