Related papers: On accumulated spectrograms

We estimate the distribution of the eigenvalues of a family of time-frequency localization operators whose eigenfunctions are the well-known Prolate Spheroidal Wave Functions from mathematical physics. These operators are fundamental to the…

We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of…

Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on $\mathbb{R}$ that have the highest concentration within a specific time interval. They are also identified as the eigenfunctions of a time-frequency…

We derive a spectral representation for the oblate spheroidal wave operator which is holomorphic in the aspherical parameter $\Omega$ in a neighborhood of the real line. For real $\Omega$, estimates are derived for all eigenvalue gaps…

Let $D_{T}$ and $B_{\Omega }$ denote the operators which cut the time content outside $T$ and the frequency content outside $\Omega $, respectively. The prolate spheroidal functions are the eigenfunctions of the operator $P_{T,\Omega…

The aim of this article is to present a time-frequency theory for orthogonal polynomials on the interval [-1,1] that runs parallel to the time-frequency analysis of bandlimited functions developed by Landau, Pollak and Slepian. For this…

We investigate an inverse problem in time-frequency localization: the approximation of the symbol of a time-frequency localization operator from partial spectral information by the method of accumulated spectrograms (the sum of the…

For time-frequency localization operators, related to the short-time Fourier transform, with symbol $R\Omega$, we work out the exact large $R$ eigenvalue behavior for rotationally invariant $\Omega$ and conjecture that the same relation…

The approximation of the eigenvalues and eigenfunctions of an elliptic operator is a key computational task in many areas of applied mathematics and computational physics. An important case, especially in quantum physics, is the computation…

We define the accumulated spectrogram associated to a locally trace class orthogonal projection operator and to a bounded set using the polar decomposition of its restriction on that set and prove a convergence theorem for accumulated…

We examine the spectrum of a family of Sturm--Liouville operators with regularly spaced delta function potentials parametrized by increasing strength. The limiting behavior of the eigenvalues under this spectral flow was described in a…

In this paper we show that the eigenfunctions can be found exactly for systems whose delay-Doppler spread function is concentrated along a straight line and they can be found in approximate sense for systems having a spread function…

Consider the Dirichlet-Laplacian in $\Omega:= (0,L)\times (0,H)$ and choose another open set $\omega\subset \Omega$. The estimate $0<C_{\omega}\leq R_{\omega}(u):=\frac{\Vert u\Vert^{2}_{L^{2}(\omega)}}{\Vert u\Vert^{2}_{L^{2}(\Omega)}}\leq…

We consider the localization of eigenfunctions for the operator $L=-\mbox{div} A \nabla + V$ on a Lipschitz domain $\Omega$ and, more generally, on manifolds with and without boundary. In earlier work, two authors of the present paper…

The eigenfunctions of the Laplacian are a central object from the realms of analytic number theory to geometric analysis. We prove that H\"ormander $L^2$-$L^{\infty}$ estimates are equivalent to restriction estimates to small geodesic…

We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in $\mathbb{R}^n$ - the eigenfunctions of the Dirichlet-to-Neumann map. Under the assumption that the domain $\Omega$ is $C^2$, we prove a…

We consider an operator function (F(\lambda)) for (\lambda\in(\sigma,\tau)\subseteq\mathbb R) whose values are semibounded selfadjoint operators in Hilbert space (\mathfrak H). Our main goal is to estimate the number (\mathcal…

In a domain $\Omega\subset \mathbb{R}^{\mathbf{N}}$ we consider a selfadjoint operator $\mathbf{T}=\mathfrak{A}^*P\mathfrak{A} ,$ where $\mathfrak{A}$ is a pseudodifferential operator of order $-l=-\mathbf{N}/2$ and $P=V\mu_{\Sigma}$ is a…

We study the approximation properties of a harmonic function $u \in H\sp{1-k}(\Omega)$, $k > 0$, on relatively compact sub-domain $A$ of $\Omega$, using the Generalized Finite Element Method. For smooth, bounded domains $\Omega$, we obtain…

In this work, we first give various explicit and local estimates of the eigenfunctions of a perturbed Jacobi differential operator. These eigenfunctions generalize the famous classical prolate spheroidal wave functions (PSWFs), founded in…