Reconstruction Of Spline Spectra-signals From Generalized Sinc Function By Finitely Many Samples

Sampling is a fundamental tool for the conversion between an analogue signal and digital signal(A/D).The classical Shannon sampling theorem shows that band-limited signal can be reconstructed by its uniform sampling and the shifted basis of the sinc function.whether the similar sampling theorem exists for non-bandlimited signals has become a topic of concern.Recently,researchers have discussed the sampling theorem of non-bandlimited signals in the space spanned by other bases.Among these bases,the basis generated by the generalized sinc function sinca is a special example.on the other hand,in some optical engineering problems,signals are often reconstructed by their Fourier samples.This paper concerns on the re-construction of spline-spectra signals in the shift-invariant space V(sinca)by finitely many Fourier samples.There are two main results on this topic:when the spectra knots of spline-spectra signals are known,the exact reconstruction formula conduct-ed 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 approx-imations to the spline-spectra signals by finitely many Fourier samples.Numerical simulations are conducted to check the conclusion of the theorem.Its main contents are as follows:Firstly,for the spline-spectra signals in shift-invariant spaces V(sinca),this pa-per concerns on the representation of spline-spectra signals that can be reconstructed by finitely many Fourier samples in the Fourier transform domain.Secondly,we reconstruct spline-spectra signals in V(sinca)by finitely many Fourier samples.Thirdly,numerical simulations of the spline spectral signal in V(sinca)are con-ducted to check the efficiency of the approximation.