# Nonlinear approximation with nonstationary Gabor frames

## TL;DR

This paper investigates the sparseness of adaptive time-frequency representations using nonstationary Gabor frames, characterizing signals with sparse expansions and providing bounds on approximation errors, supported by numerical experiments.

## Contribution

It establishes a theoretical link between sparseness and smoothness in nonstationary Gabor expansions and offers practical bounds on approximation errors.

## Key findings

- Sparse expansions correspond to smooth signals in a specific decomposition space.
- An upper bound on approximation error from thresholding coefficients is proven.
- Numerical experiments confirm the theoretical approximation rates.

## Abstract

We consider sparseness properties of adaptive time-frequency representations obtained using nonstationary Gabor frames (NSGFs). NSGFs generalize classical Gabor frames by allowing for adaptivity in either time or frequency. It is known that the concept of painless nonorthogonal expansions generalizes to the nonstationary case, providing perfect reconstruction and an FFT based implementation for compactly supported window functions sampled at a certain density. It is also known that for some signal classes, NSGFs with flexible time resolution tend to provide sparser expansions than can be obtained with classical Gabor frames. In this article we show, for the continuous case, that sparseness of a nonstationary Gabor expansion is equivalent to smoothness in an associated decomposition space. In this way we characterize signals with sparse expansions relative to NSGFs with flexible time resolution. Based on this characterization we prove an upper bound on the approximation error occurring when thresholding the coefficients of the corresponding frame expansions. We complement the theoretical results with numerical experiments, estimating the rate of approximation obtained from thresholding the coefficients of both stationary and nonstationary Gabor expansions.