B-spline approximations of the Gaussian, their Gabor frame properties, and approximately dual frames

TL;DR

This paper demonstrates that scaled B-splines can approximate Gaussian Gabor systems and their dual frames, enabling nearly perfect reconstruction with simple, explicit dual windows for sufficiently large B-spline order and small time-frequency product.

Contribution

It introduces a method to approximate Gaussian Gabor frames using scaled B-splines, providing explicit, simple dual windows for near-perfect reconstruction.

Findings

Scaled B-splines generate Gabor frames for large enough order.

Explicit approximate dual windows achieve arbitrary accuracy.

Convergence of B-splines to Gaussian in various function spaces.

Abstract

We prove that Gabor systems generated by certain scaled B-splines can be considered as perturbations of the Gabor systems generated by the Gaussian, with a deviation within an arbitrary small tolerance whenever the order $N$ of the B-spline is sufficiently large. As a consequence we show that for any choice of translation/modulation parameters $a,b>0$ with $ab<1,$ the scaled version of $B_N$ generates Gabor frames for $N$ sufficiently large. Considering the Gabor frame decomposition generated by the Gaussian and a dual window, the results lead to estimates of the deviation from perfect reconstruction that arise when the Gaussian is replaced by a scaled B-spline, or when the dual window of the Gaussian is replaced by certain explicitly given and compactly supported linear combinations of the B-splines. In particular, this leads to a family of approximate dual windows of a very simple form, leading to "almost perfect reconstruction" within any desired error tolerance whenever the product $ab$ is sufficiently small. In contrast, the known (exact) dual windows have a very complicated form. A similar analysis is sketched with the scaled B-splines replaced by certain truncations of the Gaussian. As a consequence of the approach we prove (mostly known) convergence results for the considered scaled B-splines to the Gaussian in the $L^p$-spaces, as well in the time-domain as in the frequency domain.