SAMPLING THEORY IN SIGNAL AND IMAGE PROCESSING c
Vol. 6, No. 2, May 2007, pp. 199-221
2007 SAMPLING PUBLISHING
ISSN: 1530-6429
Sinc Approximation with a Gaussian Multiplier Gerhard Schmeisser
Mathematical Institute, University of Erlangen-Nuremberg, Bismarckstrasse 1 12 D-91054 Erlangen, Germany
[email protected]
Frank Stenger
Department of Computer Science, University of Utah Salt Lake City, UT 84112 USA
[email protected]
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 nite 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 modi ed operator that this approximation method applies not only to bandlimited signals of nite energy, but also to bandlimited signals of in nite 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 N , where M and 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 Classi cation | Primary 30E10, 41A25, 41A80, 94A20; Secondary 41A30, 65B10
1
Introduction
Throughout this paper, we shall use the following notation. For 0 we denote by E the set of all function f which are entire, that is, analytic in the whole
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G. SCHMEISSER AND F. STENGER
complex plane, and satisfy lim sup r!1
log maxjzj=r jf (z )j r
:
(1)
We call E the class of entire functions of exponential type : When we have a function f : C ! C and write f 2 Lp (R); we actually mean that the restriction of f to R belongs to Lp(R): The norm in Lp(R) will be denoted by k kp for p 2 [1; 1]. The spaces Bp := E \ Lp(R) are called Bernstein spaces; see [5, x 6.1]. We have
B Bp Br B1 1
(1 p r 1):
It is well known that a function f is a signal of nite energy bandlimited to [ ; ] if and only if f is the restriction to R of a function from B2 : Introducing the sinc function by 8 < sin z if z 2 C n f0g; sinc z := : z 1 if z = 0; we may state the classical sampling theorem of Whittaker{Kotel'nikov{Shannon (see [7, p. 49]) as follows. Theorem A
Let f
2 B ; where > 0. Then, for h 2 (0; =]; we have
f (z ) =
2
1 X
1
n=
(z 2 C ):
f (hn)sinc (h 1 z n)
The series converges absolutely and uniformly on every strip K > 0.
(2)
j=zj K for any
In practice, the use of (2) is limited since the series converges slowly unless
jf (x)j decays rapidly as x ! 1: However, if we choose h strictly less than =, which is the case of oversampling, we can easily incorporate a convergence factor into the series in (2). In fact, for 0 < " < h; let 2 E" such that (0) = 1. Suppose that j(z )j decays rapidly on lines parallel to the real axis, at least as fast as O(j
