Related papers: Basisness and completeness of Fucik eigenfunctions…

We establish sufficient assumptions on sequences of Fucik eigenvalues of the one-dimensional Laplacian which guarantee that the corresponding Fucik eigenfunctions form a Riesz basis in $L^2(0,\pi)$.

We provide improved sufficient assumptions on sequences of Fucik eigenvalues of the one-dimensional Dirichlet Laplacian which guarantee that the corresponding Fucik eigenfunctions form a Riesz basis in $L^2(0,\pi)$. For that purpose, we…

The paper is concerned with the completeness property of root functions of general boundary value problems for $n \times n$ first order systems of ordinary differential equations on a finite interval. In comparison with the recent paper…

For $q>12/11$ the eigenfunctions of the non-linear eigenvalue problem associated to the one-dimensional $q$-Laplacian are known to form a Riesz basis of $L^2(0,1)$. We examine in this paper the approximation properties of this family of…

The purpose of this paper is to give a simple proof of sharp $L^\infty$ estimates for the eigenfunctions of the Dirichlet Laplacian on smooth compact Riemannian manifolds $(M,g)$ of dimension $n\ge 2$ with boundary $\partial M$ and then to…

Let $S\subset\R^d$ be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to $S$, $PW_S$, is defined to be the set of all square-integrable functions on $\R^d$ whose Fourier transforms vanish outside $S$. A…

For one-dimensional Dirac operators $$ Ly= i \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \frac{dy}{dx} + v y, \quad v= \begin{pmatrix} 0 & P \\ Q & 0 \end{pmatrix}, \;\; y=\begin{pmatrix} y_1 \\ y_2 \end{pmatrix}, $$ subject to periodic…

We find an infinite set of eigenfunctions for the Laplacian with respect to a flat metric with conical singularities and acting on degree zero bundles over special Riemann surfaces of genus greater than one. These special surfaces…

We derive several new applications of the concept of sequences of Laplacian cut-off functions on Riemannian manifolds (which we prove to exist on geodesically complete Riemannian manifolds with nonnegative Ricci curvature): In particular,…

For a partition of $[0,1]$ into intervals $I_1,\ldots,I_n$ we prove the existence of a partition of $\mathbb{Z}$ into $\Lambda_1,\ldots, \Lambda_n$ such that the complex exponential functions with frequencies in $ \Lambda_k$ form a Riesz…

Let $\lambda$ be a positive number, and let $(x_j:j\in\mathbb Z)\subset\mathbb R$ be a fixed Riesz-basis sequence, namely, $(x_j)$ is strictly increasing, and the set of functions $\{\mathbb R\ni t\mapsto e^{ix_jt}:j\in\mathbb Z\}$ is a…

In this paper we study the Riesz transform on complete and connected Riemannian manifolds $M$ with a certain spectral gap in the $L^2$ spectrum of the Laplacian. We show that on such manifolds the Riesz transform is $L^p$ bounded for all $p…

The basis functions of the Fourier interpolation formula of Radchenko and Viazovska, constructed by means of weakly holomorphic modular forms for the Hecke theta group, are entire functions of order $2$ having interesting time-frequency…

We study subspaces of Orlicz spaces $L_M$ spanned by independent copies $f_k$, $k=1,2,\dots$, of a function $f\in L_M$, $\int_0^1 f(t)\,dt=0$. Any such a subspace $H$ is isomorphic to some Orlicz sequence space $\ell_\psi$. In terms of…

The location of the spectrum and the Riesz basis property of well-posed homogeneous infinite-dimensional linear port-Hamiltonian systems on a 1D spatial domain are studied. It is shown that the Riesz basis property is equivalent to the fact…

We consider the eigenvalues of the Dirichlet and Neumann Laplacians on a bounded domain with Lipschitz boundary and prove two-term asymptotics for their Riesz means of arbitrary positive order. Moreover, when the underlying domain is…

In this paper we formulate a concrete method for determining whether a system of dilated periodic functions forms a Riesz basis in $L^2(0,1)$. This method relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20…

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and periodic or antiperiodic boundary conditions. Then using these…

The complex exponentials with integer frequencies form a basis for the space of square integrable functions on the unit interval. We analyze whether the basis property is maintained if the support of the complex exponentials is restricted…

We explore the possibilities of reaching the characterization of eigenfunction of Laplacian as a degenerate case of the inverse Paley-Wiener theorem (characterizing functions whose Fourier transform is supported on a compact annulus) for…