Bounds for the Tails of Sharp-Cutoff Filter Kernels
B. F. Logan · SIAM Journal on Mathematical Analysis · 1988
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 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.