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.
The Hilbert transform H satis es the Bedrosian identity H(fg) = fHg whenever the supports of the Fourier transforms of f, g ∈ L (ℝ) are respectively contained inAttracted by this interesting result arising from the time-frequency analysis, we investigate the existence of such an identity for a general bounded Fourier multiplier operator on L (ℝ d ) and for general support sets A and B. A geometric characterization of the support sets for the existence of the Bedrosian identity is established. Moreover, the support sets for the partial Hilbert transforms are all found. In particular, for the Hilbert transform to satisfy the Bedrosian identity, the support sets must be given as above.
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 the engineering and has received remarkable attention. It works by multiplying the sinc function in the Shannon series with a regularized Gaussian function. Recently, it was proved that the upper error bound of this method can achieve a convergence rate of the order O( 1 √ n exp(− π−δ 2 n)), where 0 < δ < π is the bandwidth and n the number of sample data. The convergence rate is by far the best convergence rate among all regularized methods for the Shannon sampling series. The main objective of this article is to present the theoretical justification and numerical verification that the convergence rate is optimal when 0 < δ < π/2 by estimating the lower error bound of the truncated Gaussian regularized Shannon sampling series.
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