Description
We present solutions for variational problems, compression and noise removal using gabor frames. For an appropriate window function, a signal $ f \in \mathcal{L}^2(\mathbb{R}^d)$ possesses a non-orthogonal gabor frames expansions in terms of the dual frames with unconditional convergence in $ \mathcal{L}^2(\mathcal(R)^d) $. We derive approximate minimizers of variational problems and compression in modulation spaces. Within the Gaussian white noise model we provide minimax bounds for rates of convergence over modulation spaces using soft-thresholding of the Gabor coefficients. Numerical experiments complement the theoretical results. Furthermore we extend our results onto $\alpha$-modulation spaces, providing a flexible Gabor-wavelet transform of signals.
