In this article, we consider the reconstruction of a bandlimited function from its finite localized sample data. Truncating the classical Shannon sampling series results in an unsatisfactory convergence rate due to the slow decay of the sinc function. To overcome this drawback, a simple and highly effective method called the Gaussian regularization of the Shannon series was proposed in engineering and has received remarkable attention. It works by multiplying the sinc function in the Shannon series with a regularized Gaussian function. L. Qian established the convergence rate of for this method, where δ<π is the bandwidth and n is the number of sample data. C. Micchelli et al. proposed a different regularized method and obtained the corresponding convergence rate of . This latter rate is by far the best among all regularized methods for the Shannon series. However, their regularized method involves the solving of a linear system and is implicit and more complicated. The main objective of this article is to show that the Gaussian regularization of the Shannon series can also achieve the same best convergence rate as that by C. Micchelli et al. We also show that the Gaussian regularization method can improve the convergence rate for the useful average sampling.

This paper characterizes the proximal operator of the piece-wise exponential function $1\!-\!e^{-|x|/\sigma }$ with a given shape parameter $\sigma \!>\!0$ , which is a popular non-convex surrogate of the $\ell _{0}$ -norm in support vector machines, compressed sensing, neural networks, etc. Although Malek-Mohammadi et al. [IEEE Transactions on Signal Processing, 64(21):5657–5671, 2016] once worked on this problem, the expressions they derived were regrettably inaccurate. In a sense, it was lacking a case. Using the Lambert W function and an extensive study of the piece-wise exponential function, we have rectified the formulation of the proximal operator of the piece-wise exponential function in light of their work. We have also undertaken a thorough analysis of this operator. A comparative analysis of eleven sparse-promoting functions in compressed sensing demonstrates the effectiveness and efficiency of the piece-wise exponential function.