Convergence Analysis of the Gaussian Regularized Shannon Sampling Series

Abstract

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 decayness 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 regularization Gaussian function. L. Qian (Proc. Amer. Math. Soc., 2003) established the convergence rate of O( √ n exp(− π−δ 2 n)) for this method, where δ < π is the bandwidth and n is the number of sample data. C. Micchelli et al. (J. Complexity, 2009) proposed a different regularization method and obtained the corresponding convergence rate of O( 1 √ n exp(− π−δ 2 n)). This latter rate is by far the best among all regularization methods for the Shannon series. However, their regularized function involves the solving of a linear system and is implicit and more complicated. The main objective of this note is to show that the Gaussian regularized 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. Numerical experiments are presented to justify the obtained results.

On multidimensional sinc-Gauss sampling formulas for analytic functions

“…The theoretical error analysis is established via a complex analytic approach, and the convergence rate is of exponential order. Furthermore, this paper extends the work of Lin and Zhang in [10] to the complex domains. We can use the multivariate kernel function to extend the two-dimensional Hermite-Gauss sampling formula, introduced in [3,6], to the general multidimensional case.…”

“…) by a bound of exponential order. In other word, if f ∈ B 2 σ (R n ), then for x ∈ (0, 1) n , Lin and Zhang have shown in [10,Theorem 2.5] that σ (R n ) and needs an adaptation when we approximate the function f outside the region (0, 1) n . Here we would like to point out that the modification of the sampling series with Gaussian multipliers, which uses samples only from the function itself, is called sinc-Gauss formula; cf., e.g., [4,19,21].…”

“…[4,7]. To the best of our knowledge, there is no single study on the modification of the general multivariate sampling series with Gaussian multipliers, the work of Lin and Zhang in [10] being an exception. They have studied their modification based on a Fourier-analytic approach.…”

Gaussian Regularized Periodic Nonuniform Sampling Series

“…e convergence order [18,[25][26][27] of Gaussian regularization of the Shannon sampling series is the best among other known regularization methods [28][29][30][31][32] because of the good time-frequency concentration of the Gaussian function. In this paper, we proposed the Gaussian regularized periodic nonuniform sampling series and proved that this series is strictly exponentially decaying.…”

An Optimal Convergence Rate for the Gaussian Regularized Shannon Sampling Series

“…The latter rate (1.2) is by far the best convergence rate among all regularized methods [7,10,12,15] for the Shannon sampling series. Due to its simplicity and high accuracy, the Gaussian regularized Shannon sampling series has been widely applied to scientific and engineering computations.…”