Related papers: Compactly supported Gabor orthonormal bases

We discuss `hd-compactifications' of $\SL(2,\bbK)$ for $\bbK=\bbC$ or $\bbR.$ These are compact manifolds with boundary on which both the Schwartz and the Harish-Chandra Schwartz spaces are shown to be relatively standard spaces of conormal…

In this work we investigate the question of constructions of the possible Fourier bases $E(\Lambda)=\{e^{2\pi i \lambda x}:\lambda\in\Lambda\}$ for the Hilbert space $L^2(\mu_4)$, where $\mu_4$ is the standard middle-fourth Cantor measure…

A ladder algebraic structure for $L^2(\mathbb{R}^+)$ which closes the Lie algebra $h(1)\oplus h(1)$, where $h(1)$ is the Heisenberg-Weyl algebra, is presented in terms of a basis of associated Laguerre polynomials. Using the Schwinger…

We construct a uniformly discrete sequence $\{\lambda_1 < \lambda_2 < \cdots\} \subset \mathbb{R}$ and functions $g$ and $\{g_n^*\}$ in $L^2(\mathbb{R})$, such that every $f \in L^2(\mathbb{R})$ admits a series expansion \[ f(x) =…

Orthogonal polynomials on the product domain $[a_1,b_1] \times [a_2,b_2]$ with respect to the inner product $$ \langle f,g \rangle_S = \int_{a_1}^{b_1} \int_{a_2}^{b_2} \nabla f(x,y)\cdot \nabla g(x,y)\, w_1(x)w_2(y) \,dx\, dy + \lambda…

A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard…

It is well-known that the functions $f \in L^1(\mathbb{R}^d)$ whose translates along a lattice $\Lambda$ form a tiling, can be completely characterized in terms of the zero set of their Fourier transform. We construct an example of a…

We study regularity results for nonlinear parabolic systems of $p$-Laplacian type with inhomogeneous boundary and initial data, with $p\in(\frac{2n}{n+2},\infty)$. We show bounds on the gradient of solutions in the Lebesgue-spaces with…

We show that a Lagrangian inclusion in $\mathbb C^2$ with double transverse self-intersection points and standard open Whitney umbrellas is rationally convex. As an application we show that any compact surface $S$, except $S^2$ and $\mathbb…

We introduce the concept of gauged Lagrangian $1$-forms, extending the notion of Lagrangian $1$-forms to the setting of gauge theories. This general formalism is applied to a natural geometric Lagrangian $1$-form on the cotangent bundle of…

Let $I=(a,b)\times(c,d)\subset {\mathbb R}_{+}^2$ be an index set and let $\{G_{\alpha}(x) \}_{\alpha \in I}$ be a collection of Gaussian functions, i.e. $G_{\alpha}(x) = \exp(-\alpha_1 x_1^2 - \alpha_2 x_2^2)$, where $\alpha = (\alpha_1,…

The frame set of a window $\phi\in L^2(\mathbb{R})$ is the subset of all lattice parameters $(\alpha, \beta)\in \mathbb{R}^2_+$ such that $\mathcal{G}(\phi,\alpha,\beta)=\{e^{2\pi i\beta m\cdot}\phi(\cdot-\alpha k) : k, m\in\mathbb{Z}\}$…

We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a…

Given a simple Lie group $G$, we show that the lattices in $G$ are weakly uniformly discrete. This is a strengthening of the Kazhdan-Margulis theorem. Our proof however is straightforward --- considering general IRS rather than lattices…

In this paper Gabor system of certain type based on the unitary dual of the Heisenberg group $\mathbb{H}^n$ is introduced and a sufficient condition is obtained for the Gabor system to be a Bessel sequence for…

Let a sequence $(P_n)$ of polynomials in one complex variable satisfy a recurre ce relation with length growing slowlier than linearly. It is shown that $(P_n) $ is an orthonormal basis in $L^2_{\mu}$ for some measure $\mu$ on $\C$, if and…

Several results in functional analysis are extended to the setting of $L^0$-modules, where $L^0$ denotes the ring of all measurable functions $x\colon \Omega\to \mathbb{R}$. The focus is on results involving compactness. To this end, a…

The gauged sigma model with target $\mathbb{P}^1$, defined on a Riemann surface $\Sigma$, supports static solutions in which $k_+$ vortices coexist in stable equilibrium with $k_-$ antivortices. Their moduli space is a noncompact complex…

We prove universal lower bounds for discrepancies (i.e. sizes of spectral gaps of averaging operators) of measure-preserving actions of a locally compact group on probability spaces. For example, a locally compact Hausdorff unimodular group…

We study $\lambda$-hypersurfaces that are critical points of a Gaussian weighted area functional $\int_{\Sigma} e^{-\frac{|x|^2}{4}}dA$ for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems…