Andrei Caragea

Katholische Universität Eichstätt-Ingolstadt (KU) | KU · mathematics

Gabor systems are used in fields ranging from audio processing to digital communication. Such a Gabor system (g,Λ) consists of all time-frequency shifts π(λ)g of a window function g∈L2(R) along a lattice Λ⊂R2. We focus on Gabor systems that are also Riesz sequences, meaning that one can stably reconstruct the coefficients c=(cλ)λ∈Λ from the functio...

We prove that if $$I_\ell = [a_\ell ,b_\ell )$$ I ℓ = [ a ℓ , b ℓ ) , $$\ell =1,\ldots ,L$$ ℓ = 1 , … , L , are disjoint intervals in [0, 1) with the property that the numbers $$1, a_1, \ldots , a_L, b_1, \ldots , b_L$$ 1 , a 1 , … , a L , b 1 , … , b L are linearly independent over $${\mathbb {Q}}$$ Q , then there exist pairwise disjoint sets $$\L...

We prove that if $I_\ell$, $\ell=1, \ldots, L$, are disjoint separated intervals in $[0,1)$, then there exist pairwise disjoint sets $\Lambda_\ell \subset \mathbb{Z}$, $\ell=1, \ldots, L$, such that for every $J \subset \{ 1, \ldots , L \}$, the system $\{e^{2\pi i \lambda x} : \lambda\in \cup_{\ell \in J} \, \Lambda_\ell \}$ is a Riesz basis for $...

Let G⊂L2(R) be the subspace spanned by a Gabor Riesz sequence (g,Λ) with g∈L2(R) and a lattice Λ⊂R2 of rational density. It was shown recently that if g is well-localized both in time and frequency, then G cannot contain any time-frequency shift π(z)g of g with z∈R2∖Λ. In this paper, we improve the result to the quantitative statement that the L2-d...

We show that complex-valued neural networks with the modReLU activation function $\sigma(z) = \mathrm{ReLU}(|z| - 1) \cdot z / |z|$ can uniformly approximate complex-valued functions of regularity $C^n$ on compact subsets of $\mathbb{C}^d$, giving explicit bounds on the approximation rate.

We prove bounds for the approximation and estimation of certain classification functions using ReLU neural networks. Our estimation bounds provide a priori performance guarantees for empirical risk minimization using networks of a suitable size, depending on the number of training samples available. The obtained approximation and estimation rates a...

We extend the Balian–Low theorem to Gabor subspaces of L2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2({\mathbb {R}})$$\end{document} by involving the concept...

We consider non-complete Gabor frame sequences generated by an $S_0$-function and a lattice $\Lambda$ and prove that there is $m \in \mathbb{N}$ such that all time-frequency shifts leaving the corresponding Gabor space invariant have their parameters in $\tfrac{1}{m}\Lambda$. We also investigate time-frequency shift invariance under duality aspects...

Let $\mathcal G \subset L^2(\mathbb R)$ be the subspace spanned by a Gabor Riesz sequence $(g,\Lambda)$ with $g \in L^2(\mathbb R)$ and a lattice $\Lambda \subset \mathbb R^2$ of rational density. It was shown recently that if $g$ is well-localized both in time and frequency, then $\mathcal G$ cannot contain any time-frequency shift $\pi(z) g$ of $...