Riesz Basis In L2 Research Articles

Finite Gabor systems and uncertainty principle for block sliding discrete fourier transform

In this paper, we study the finite Gabor system for oversampling schemes. A characterization of dual finite Gabor tight frame using discrete time Zak transform is given. Also, a method to calculate the coefficients of the finite Gabor system expansion in the case of oversampling and a necessary and sufficient condition for the existence of biorthogonal pair of Riesz basis in l2(ZL) is given. Further, we introduce the notion of block sliding discrete Fourier transform (BSDFT) which reduces the computational complexity and give uncertainty principle for BSDFT. An uncertainty principle for two finite Parseval Gabor frames in terms of sparse representations is given. Finally, using the notion of numerical sparsity, an uncertainty principle for finite Gabor frames is given.

Bases of complex exponentials with restricted supports

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 to possibly overlapping subsets of the unit interval. We show, for example, that if S1,…,SK⊂[0,1] are finite unions of intervals with rational endpoints that cover the unit interval, then there exists a partition of Z into sets Λ1,…,ΛK such that ⋃k=1K{e2πiλ(⋅)χSk:λ∈Λk} is a Riesz basis for L2[0,1]. Here, χS denotes the characteristic function of S.

Basis properties of Fučík eigenfunctions for the Neumann Laplacian

We investigate the basis properties of sequences of Fučík eigenfunctions of the one-dimensional Neumann Laplacian. We show that any such sequence is complete in L2(0,π) and a Riesz basis in the subspace of functions with zero mean. Moreover, we provide sufficient assumptions on Fučík eigenvalues which guarantee that the corresponding Fučík eigenfunctions form a Riesz basis in L2(0,π) and we explicitly describe the corresponding biorthogonal system.

Boundary null-controllability of two coupled parabolic equations: simultaneous condensation of eigenvalues and eigenfunctions

Let the matrix operator L = D∂xx + q (x) A0, with D = diag (1, ν), ν ≠ 1, q ∈ L∞ (0, π), and A0 is a Jordan block of order 1. We analyze the boundary null controllability for the system yt − Ly = 0. When [see formula in PDF] and q is constant, q = 1 for instance, there exists a family of root vectors of [see formula in PDF] forming a Riesz basis of L2(0,π;ℝ2). Moreover F. Ammar Khodja et al. [J. Funct. Anal. 267 (2014) 2077–2151] shows the existence of a minimal time of control depending on condensation of eigenvalues of [see formula in PDF], that is to say the existence of T0 (ν) such that the system is null controllable at time T > T0 (ν) and not null controllable at time T < T0 (ν). In the same paper, the authors prove that for all τ ∈ [0, + ∞], there exists ν ∈] 0, + ∞ [ such that T0 (ν) = τ. When q depends on x, the property of Riesz basis is no more guaranteed. This leads to a new phenomena: simultaneous condensation of eigenvalues and eigenfunctions. This condensation affects the time of null controllability.

Riesz basis of exponential family for a hyperbolic system

Abstract This paper studies a linear hyperbolic system with boundary conditions that was first studied under some weaker conditions in [8, 11]. Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. It is shown that the associated linear system is the infinitesimal generator of a C 0 {C_{0}} -semigroup; its spectrum consists of zeros of a sine-type function, and its exponential system { e λ n ⁢ t } n ≥ 1 {\{e^{\lambda_{n}t}\}_{n\geq 1}} constitutes a Riesz basis in L 2 ⁢ [ 0 , T ] {L^{2}[0,T]} . Furthermore, by the spectral analysis method, it is also shown that the linear system has a sequence of eigenvectors, which form a Riesz basis in Hilbert space, and hence the spectrum-determined growth condition is deduced.

On the Riesz basis property of root vectors system for 2 × 2 Dirac type operators

The paper is concerned with the Riesz basis property of a boundary value problem associated in L2[0,1]⊗C2 with the following 2×2 Dirac type equation(0.1)Ly=−iB−1y′+Q(x)y=λy,B=(b100b2),y=col(y1,y2), with a summable potential matrix Q∈L1[0,1]⊗C2×2 and b1<0<b2. If b2=−b1=1 this equation is equivalent to one dimensional Dirac equation. It is proved that the system of root functions of a linear boundary value problem constitutes a Riesz basis in L2[0,1]⊗C2 provided that the boundary conditions are strictly regular. By analogy with the case of ordinary differential equations, boundary conditions are called strictly regular if the eigenvalues of the corresponding unperturbed (Q=0) operator are asymptotically simple and separated. In opposite to the Dirac case there is no simple algebraic criterion of the strict regularity whenever b1+b2≠0. However under certain restrictions on coefficients of the boundary linear forms we present certain algebraic criteria of the strict regularity in the latter case. In particular, it is shown that regular separated boundary conditions are always strictly regular while antiperiodic boundary conditions are strictly regular if b1, b2 are coprime integers of different parity. The proof of the main result is based on existence of triangular transformation operators for system (0.1). Their existence is also established here in the case of a summable Q. In the case of regular (but not strictly regular) boundary conditions we prove the Riesz basis property with parentheses. The main results are applied to establish the Riesz basis property of the dynamic generator of spatially non-homogeneous damped Timoshenko beam model.

Riesz basis of wavelets constructed from trigonometric B-splines

In this paper, we construct a class of compactly supported wavelets by taking trigonometric B-splines as the scaling function. The duals of these wavelets are also constructed. With the help of these duals, we show that the collection of dilations and translations of such a wavelet forms a Riesz basis of 𝕃2(ℝ). Moreover, when a particular differential operator is applied to the wavelet, it also generates a Riesz basis for a particular generalized Sobolev space. Most of the proofs are based on three assumptions which are mild generalizations of three important lemmas of Jia et al. [Compactly supported wavelet bases for Sobolev spaces, Appl. Comput. Harmon. Anal. 15 (2003) 224–241].

Approximation rates for interpolation of Sobolev functions via Gaussians and allied functions

A Riesz-basis sequence forL2[−π,π] is a strictly increasing sequence X:=(xj)j∈Z in R such that the set of functions (e−ixj(⋅))j∈Z is a Riesz basis for L2[−π,π]. Given such a sequence and a parameter 0<h≤1, we consider interpolation of functions g∈W2k(R) at the set (hxj)j∈Z via translates of the Gaussian kernel. Existence is shown of an interpolant of the form IhX(g)(x):=∑j∈Zaje−(x−hxj)2,x∈R, which is continuous and square-integrable on R, and satisfies the interpolatory condition IhX(g)(hxj)=g(hxj),j∈Z. Moreover, the use of the parameter h gives approximation rates of order hk. Namely, there is a constant independent of g such that ‖IhX(g)−g‖L2(R)≤Chk∣g∣W2k(R). Interpolation using translates of certain functions other than the Gaussian, so-called regular interpolators, is also considered and shown to exhibit the same approximation rates.

STUDY BASICITY OF ROOT FUNCTIONS OF THE SCHRODINGER OPERATOR WITH A NON-LOCAL PERTURBATION

The asymptotics of the eigenvalues is received, and conclusions about the stability and instability of the basis property of the system of eigen- functions and associated functions of the Schrodinger operator for various oc- casions of the disarmed regularity of the boundary conditions are done. In the paper there is considered the Samarsky - Ionkin spectral problem for the Schrodinger equation with an integral perturbation in the boundary conditions. It is assumed that the unperturbed problem has a system of eigenfunctions forming a Riesz basis in L2(0,1) . It is shown that the basis property of the systems of root functions of a problem can be varied under any arbitrarily small variation of the kernel of the integral perturbation.

On a Riesz basis of exponentials related to the eigenvalues of an analytic operator and application to a non-selfadjoint problem deduced from a perturbation method for sound radiation

In the present paper, we prove that the family of exponentials associated to the eigenvalues of the perturbed operator T(ɛ) ≔ T0 + ɛT1 + ɛ2T2 + … + ɛkTk + … forms a Riesz basis in L2(0, T), T > 0, where ɛ∈C, T0 is a closed densely defined linear operator on a separable Hilbert space H with domain D(T0) having isolated eigenvalues with multiplicity one, while T1, T2, … are linear operators on H having the same domain D⊃D(T0) and satisfying a specific growing inequality. After that, we generalize this result using a H-Lipschitz function. As application, we consider a non-selfadjoint problem deduced from a perturbation method for sound radiation.

On the vaguelet and Riesz properties of [formula omitted]-unbounded transformations of orthogonal wavelet bases

In this work, we prove that certain L2-unbounded transformations of orthogonal wavelet bases generate vaguelets. The L2-unbounded functions involved in the transformations are assumed to be quasi-homogeneous at high frequencies. We provide natural examples of functions which are not quasi-homogeneous and for which the resulting transformations are not vaguelets. We also address the related question of whether the considered family of functions is a Riesz basis in L2(R). The Riesz property could be deduced directly from the results available in the literature or, as we outline, by using the vaguelet property in the context of this work. The considered families of functions arise in wavelet-based decompositions of stochastic processes with uncorrelated coefficients.

Weyl-Heisenberg Riesz Bases Generated by Two Intervals

Let {e2πimxg(x−n)}m,n∈ℤ be a Weyl-Heisenberg system and let g be the characteristic function of a union of two separated intervals. We systematically study the window function g such that the Weyl-Heisenberg system consists of a Riesz basis of L2(ℝ).

Construction of a class of multivariate compactly supported wavelet bases for L 2(ℝ d )

In this paper, for a given d×d expansive matrix M with |detM| = 2, we investigate the compactly supported M-wavelets for L2(ℝd). Starting with a pair of compactly supported refinable functions ϕ and \(\tilde \varphi \) satisfying a mild condition, we obtain an explicit construction of a compactly supported wavelet ψ such that {2j/2ψ(Mj · −k): j ∈ ℤ, k ∈ ℤd} forms a Riesz basis for L2(ℝd). The (anti-)symmetry of such ψ is studied, and some examples are also provided.

Exponential Riesz Bases, Discrepancy of Irrational Rotations and BMO

We study the basis property of systems of exponentials with frequencies belonging to ‘simple quasicrystals’. We show that a diophantine condition is necessary and sufficient for such a system to be a Riesz basis in L 2 on a finite union of intervals. For the proof we extend to BMO a theorem of Kesten about the discrepancy of irrational rotations of the circle.

Approximation properties of the q -sine bases

For , the eigenfunctions of the nonlinear eigenvalue problem associated with 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 functions and its dual, and establish a non-orthogonal spectral method for the p -poisson boundary value problem and its corresponding parabolic time-evolution initial value problem with stochastic forcing. The principal objective of our analysis is the determination of optimal values of q for which the best approximation is achieved for a given p problem.

Nonuniform Sampling and Recovery of Multidimensional Bandlimited Functions by Gaussian Radial-Basis Functions

Let S⊂ℝd be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to S, PWS, is defined to be the set of all square-integrable functions on ℝd whose Fourier transforms vanish outside S. A sequence (xj:j∈ℕ) in ℝd is said to be a Riesz-basis sequence for L2(S) (equivalently, a complete interpolating sequence for PWS) if the sequence \((e^{-i\langle x_{j},\cdot \rangle }:j\in \mathbb {N})\) of exponential functions forms a Riesz basis for L2(S). Let (xj:j∈ℕ) be a Riesz-basis sequence for L2(S). Given λ>0 and f∈PWS, there is a unique sequence (aj) in l2 such that the function $$I_\lambda(f)(x):=\sum_{j\in \mathbb {N}}a_je^{-\lambda \|x-x_j\|_2^2},\quad x\in \mathbb {R}^d,$$ is continuous and square integrable on ℝd, and satisfies the condition Iλ(f)(xn)=f(xn) for every n∈ℕ. This paper studies the convergence of the interpolant Iλ(f) as λ tends to zero, i.e., as the variance of the underlying Gaussian tends to infinity. The following result is obtained: Let \(\delta\in(\sqrt{2/3},1]\) and \(0<\beta<\sqrt{3\delta^{2}-2}\). Suppose that δB2⊂Z⊂B2, and let (xj:j∈ℕ) be a Riesz basis sequence for L2(Z). If \(f\in PW_{\beta B_{2}}\), then \(f=\lim_{\lambda\to 0^{+}}I_{\lambda}(f)\) in L2(ℝd) and uniformly on ℝd. If δ=1, then one may take β to be 1 as well, and this reduces to a known theorem in the univariate case. However, if d≥2, it is not known whether L2(B2) admits a Riesz-basis sequence. On the other hand, in the case when δ<1, there do exist bodies Z satisfying the hypotheses of the theorem (in any space dimension).

On a nonlocal perturbation of a periodic eigenvalue problem

It is well known that the system of eigenfunctions of a formally self-adjoint differential operator with arbitrary self-adjoint boundary conditions providing a point spectrum is an orthonormal basis in the space L2. The problem as to whether the basis property is preserved under a weak, in a sense, perturbation of the original self-adjoint operator with discrete spectrum was considered in numerous papers (e.g., see [1, 2]). For a nonself-adjoint operator whose system of root functions is a Riesz basis in L2, a similar problem was investigated in [3]. In the present paper, on the interval [0, 1], we consider the problem

On the riesz basisness of the root functions of the nonself-adjoint sturm-liouville operator

In this article we obtain the asymptotic formulas for eigenfunctions and eigenvalues of the nonself-adjoint Sturm-Liouville operators with periodic and antiperiodic boundary conditions, when the potential is an arbitrary summable complex-valued function. Then using these asymptotic formulas, we find the conditions on Fourier coefficients of the potential for which the eigenfunctions and associated functions of these operators form a Riesz basis inL 2(0, 1).

The Spectral Expansion for a Nonself-adjoint Hill Operator with a Locally Integrable Potential

We construct the spectral expansion for the one-dimensional Schrödinger operatorL=−d2dx2+q(x),−∞<x<∞,in L2(−∞,∞), where q(x) is a 1-periodic, Lebesgue integrable on [0,1], and complex-valued potential. We obtain the asymptotic formulas for the eigenfunctions and eigenvalues of the operator Lt, for t≠0, π, generated by this operation in L2[0,1] and the t-periodic boundary conditions. Using it, we prove that the eigenfunctions and associated functions of Lt form a Riesz basis in L2[0,1] for t≠0, π. Then we find the spectral expansion for the operator L.

Perturbation of orthonormal bases inL 2-spaces

The paper improves and generalizes a classical result from Paley and Wiener in their book on Fourier transforms. Paley and Wiener gave conditions on functionsh n that imply that the sequence (1+h n (x))e inx is a Riesz basis forL 2[−π,π]. These conditions involve theL 2-norm of the second derivativesh ″ . The new results replace the differential operatory→y″ by more general differential operators inL 2-spaces, in particular, by the Hermite differential operator inL 2(R), ande inx by arbitrary orthonormal bases.