Frames by Multiplication

CABRELLI, CARLOS A.; HEINEKEN, SIGRID; MOLTER, URSULA M.; BALASZ, PETER

Current Development in Theory and Applications of Wavelets

Pushpa Publishing House

Lugar: Allahabad; Año: 2011 vol. 5 p. 165 - 186

0973-5607

In this note we study properties of a set of irregular translates of a function in $L^2(RR^d)$. This is achieved by looking at a set of exponentials restricted to a set $E subset RR^d$ with frequencies in a countable set $Lambda$. The results are obtained by analyzing which properties of this set of exponentials are preserved when multiplied by the Fourier transform of a function $h in L^2(E)$. This in turn gives information on the set of $Lambda$-translates of $h$. In particular we study frame and Riesz basis properties. Using density results due to Beurling, we prove the existence and give ways to construct frames by irregular translates.