Frames and Riesz bases of irregular translates

SIGRID HEINEKEN

Muscat, Oman

Conferencia; International Conference on Analysis and applications; 2010

A sequence ${f_k}_{kinmathbb{Z}}$ is a {sl frame} for a separableHilbert space $mathcal{H}$ if there exist positive constants $A$and $B$ that satisfy $A|f|^2leqsum_{kinmathbb{Z}}|langle f,f_k angle|^2 leqB|f|^2$ for all $f in mathcal{H}.$ In this work we study frames and Riesz bases obtained by translating a function along an irregular set of points. Let $PW_E$ be the space of a function in $L^2(mathbb{R^d})$ whose Fourier transform is supported in $E.$ We find conditions for the familiy of irregular translates of a function to be a frame for $PW_E.$ For this, we investigate systems of the form ${he_lambda}_{lambdainLambda}$ where $hin L^2(E)$ and ${e_lambda}_{lambdainLambda}$ are complex exponentials in $L^2(E),$ and also a certain Gramian function. These results have possible applications in image and digital data transmission, speech coding, general signal processing and geophysics.