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 \(V(\hbox {sinc}_{a})\) by finitely many Fourier samples, where \(\hbox {sinc}_{a}\) 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.

在许多数学/工程问题（例如反Radon变换和光学衍射层析成像）中，研究了通过傅立叶（变换）样本重建信号的过程。本文关注有限数量的傅立叶样本对\（V（\ hbox {sinc} _ {a}）\）中样条谱信号的重构，其中\（\ hbox {sinc} _ {a} \）是广义的Sinc函数。关于此主题有两个主要结果。当知道谱的结点时，在第一个主定理中建立了由有限多个傅立叶样本进行的精确重建公式。当光谱结未知时，在第二个主要定理中，我们也通过有限多个傅立叶样本建立了样条光谱信号的近似值。进行数值模拟以检查近似效率。