Gabor frames with trigonometric spline dual windows

Abstract

A Gabor system is a collection of modulated and translated copies of a window function. If we have a signal in $L^2(\mathbb{R})$, it can be analyzed with a Gabor system generated by a certain window $g$ and then synthesized with a Gabor system generated by another window $h$. If this leads us to a perfect reconstruction, we say that $g$ and $h$ are dual Gabor windows.

Few explicit examples of dual window pairs are known. This thesis constructs explicit examples of Gabor dual windows with trigonometric form. The windows have fixed support and have an arbitrary smoothness. Also, in the discrete time domain, the trigonometric form allows us to evaluate the Gabor coefficients efficiently using the Discrete Fourier Transform. For the higher dimensional cases, we find window examples for a large class of modulation parameter lattices, including shear lattices. Also, a sufficient condition on the norm of the modulation lattice to have explicit dual Gabor windows is presented, for every dimension.