Spectral properties of a Schrödinger equation with a class of complex potentials and a general boundary condition

In this paper we investigate the spectrum and the spectral singularities of an operator L generalized in by the differential expression

     

Spectral analysis of a family of second-order elliptic operators with nonlocal boundary condition indexed by a probability measure

Let D⊂Rd be a bounded domain and let

     

Singular Sturm-Liouville problems whose coefficients depend rationally on the eigenvalue parameter

Let −Dω(·,z)D+q be a differential operator in L²(0,∞) whose leading coefficient contains the eigenvalue parameter z. For the case that ω(·,z) has the particular form

     

Convergence of cascade algorithms by frequency approach

Starting with an initial function ?₀, the cascade algorithm generates a sequence by cascade operator Qa defined by

     

Functional calculi for convolution operators on a discrete, periodic, solvable group

Suppose T is a bounded self-adjoint operator on the Hilbert space L²(X,μ) and let

     

Derivatives of the L-cosine transform

The L^p-cosine transform of an even, continuous function is defined by

     

L stability estimate for a one-dimensional Boltzmann equation with inelastic collisions

In this paper, we study the L¹ stability of a one-dimensional Boltzmann equation on the line with inelastic collisions in Rend. Sem. Mat. Fis. Milano 67 (1997) 169-179. Under the suitable assumptions on the initial data, we construct a nonlinear functional which measures L¹ distance between two mild solutions, and is nonincreasing in time t. Using the time-decay estimate of , we show that mild solutions are L¹-stable:

     

Refinement type equations and Grincevi?jus series

We consider L¹-solutions of the following refinement type equations

     

Asymptotic properties of generalized Laguerre orthogonal polynomials

In the present paper we deal with the polynomials L_n^(α,M,N) (x) orthogonal with respect to the Sobolev inner product

     

Inverse spectral problems for Bessel operators

We study the inverse spectral problem for a class of Bessel operators given in L₂(0,1) by the differential expression

     

Heisenberg inequalities for Jacobi transforms

The classical Heisenberg uncertainty principle states that for f∈L²(R),

     

On a filter for exponentially localized kernels based on Jacobi polynomials

Let , and for k=0,1,…, denote the orthonormalized Jacobi polynomial of degree k. We discuss the construction of a matrix H so that there exist positive constants c, c₁, depending only on H, α, and β such that

Specializing to the case of Chebyshev polynomials, , we apply this theory to obtain a construction of an exponentially localized polynomial basis for the corresponding L² space.     

The principle of general localization on unit sphere

In this paper we study the general localization principle for Fourier-Laplace series on unit sphere SN⊂RN+1. Weak type (1,1) property of maximal functions is used to establish the estimates of the maximal operators of Riesz means at critical index . The properties Jacobi polynomials are used in estimating the maximal operators of spectral expansions in L₂(SN). For extending positive results on critical line , 1?p?2, we apply interpolation theorem for the family of the linear operators of weak types. The generalized localization principle is established by the analysis of spectral expansions in L₂. We have proved the sufficient conditions for the almost everywhere convergence of Fourier-Laplace series by Riesz means on the critical line.     

On the regularity of the Navier-Stokes equation in a thin periodic domain

We address the global regularity of solutions of the Navier-Stokes equations in a thin domain Ω=[0,L₁]×[0,L₂]×[0,?] with periodic boundary conditions, where L₁,L₂>0 and ?∈(0,1/2). We prove that if