Bounds for the Tails of Sharp-Cutoff Filter Kernels

In communication theory, it is convenient to deal with bandlimited signals obtained by convolving an arbitrary bounded function with a filter kernel $k(t;\alpha ,\beta )$ whose Fourier transform is 1 over the interval $( - \alpha ,\alpha )$, and vanishes outside the interval $( - \beta ,\beta )$, $0...

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Veröffentlicht in:SIAM journal on mathematical analysis 1988-03, Vol.19 (2), p.372-376
1. Verfasser: Logan, B. F.
Format: Artikel
Sprache:eng
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Zusammenfassung:In communication theory, it is convenient to deal with bandlimited signals obtained by convolving an arbitrary bounded function with a filter kernel $k(t;\alpha ,\beta )$ whose Fourier transform is 1 over the interval $( - \alpha ,\alpha )$, and vanishes outside the interval $( - \beta ,\beta )$, $0 < \alpha < \beta < \infty $. For ${\alpha / \beta }$ near 1, the sharp-cutoff case, the $L_1 $-norm of the kernel must be large. In this paper, estimates are given for $\int_{| t | > T} {| {k(t;\alpha ,\beta )} |dt} $, the norm in the tails of the kernel, which show that $T$ must grow like $(\beta - \alpha )^{ - 1} $ as $\alpha \to \beta $ in order for the norm in the tails to be (say) less than 1. This result confirms a conjecture of J. C. Lagarias and A. M. Odlyzko who used such filter kernels in a method for computing $\pi (x)$, the number of primes not exceeding $x$.
ISSN:0036-1410
1095-7154
DOI:10.1137/0519027