ZEROS OF FINITE WAVELET SUMS

Abstract

For certain analytic functions $\psi$, a lower Riesz bound for a finite wavelet system generated by $\psi$, yields an upper bound for the number of zeros on a bounded interval of the corresponding wavelet sums. In particular, we show that if the Fourier transform of $\psi$ is compactly supported, say on $[-\Omega,\Omega]$, and if $B \gt 2e \Omega$, then any finite sum $\sum_{|k| \leq \alpha/2} a_{k} \psi(x-k)$ cannot have more than $B \alpha$ zeros in $[-\alpha,\alpha]$ for $\alpha \gt 0$ sufficiently large.