Extract
Sinc approximation with a Gaussian multiplier.
Abstract
Recently, it was shown with the help of Fourier analysis that by incorporating a Gaussian multiplier into the truncated classical sampling series, one can approximate bandlimited signals of finite energy with an error that decays exponentially as a function of the number of involved samples. Based on complex analysis, we show for a slightly modified operator that this approximation method applies not only to bandlimited signals of finite energy, but also to bandlimited signals of infinite energy, to classes of nonbandlimited signals, to all entire functions of exponential type (including those whose samples increase exponentially), and to functions analytic in a strip and not necessarily bounded. Moreover, the method extends to nonreal argument. In each of these cases, the use of 2N + 1 samples results in an error bound of the form M[e.sup.-[alpha]N], where M and [alpha] are positive numbers that do not depend on N. The power of the method is illustrated by several examples. Key words and phrases: sinc approximation, sampling series, Gaussian convergence factor, error bounds, entire functions of exponential type, functions analytic in a strip 2000 AMS Mathematics Subject Classification--Primary 30E10, 41A25, 41A80, 94A20; Secondary 41A30, 65B10 1 Introduction Throughout this paper, we shall use the following notation. For [sigma] [greater than or equal to] 0 we denote by [[??].sub.[sigma]] the set of all function f which are entire, that is, analytic in the whole complex plane, and satisfy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1) We call [[??].sub.[sigma]] the class of entire functions of exponential type [sigma]. When we have a function f : [??] [right arrow] [??] and write f [member of] [L.sup.p]([??]), we actually mean that the r...See the full content of this document
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