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

     

Sturm-Liouville operators and their spectral functions

Assume that the differential operator −DpD+q in L²(0,∞) has 0 as a regular point and that the limit-point case prevails at ∞. If p≡1 and q satisfies some smoothness conditions, it was proved by Gelfand and Levitan that the spectral functions σ(t) for the Sturm-Liouville operator corresponding to the boundary conditions (pu′)(0)=τu(0), , satisfy the integrability condition . The boundary condition u(0)=0 is exceptional, since the corresponding spectral function does not satisfy such an integrability condition. In fact, this situation gives an example of a differential operator for which one can construct an analog of the Friedrichs extension, even though the underlying minimal operator is not semibounded. In the present paper it is shown with simple arguments and under mild conditions on the coefficients p and q, including the case p≡1, that there exists an analog of the Friedrichs extension for nonsemibounded second order differential operators of the form −DpD+q by establishing the above mentioned integrability conditions for the underlying spectral functions.     

Left-definite spaces of singular Sturm-Liouville problems

The left-definite Hilbert spaces for singular Sturm-Liouville problems of the limit-circle type are explicitly constructed. The construction only uses an explicit form of the left-definite boundary conditions, together with a principle solution and a non-principle solution of the differential equation involved.     

An inversion theorem for integral transforms related to singular Sturm-Liouville problems on the half line

We extend the classical theory of singular Sturm-Liouville boundary value problems on the half line, as developed by Titshmarsh and Levitan to generalized functions in order to obtain a general approach to handle many integral transforms, such as the sine, cosine, Weber, Hankel, and the K-transforms, in a unified way. This approach will lead to an inversion formula that holds in the sense of generalized functions. More precisely, for [0,) and 0<, let (x,) be a solution of the Sturm-Liouville equation

扇形和角形上的Nevanlinna类及其积分表示或分解原理

In this paper we give some sufficient conditions for analytic functions which are not identically zero and belong to Nevanlinna class in the sector and angular domain. Moreover, their integral expressions or factorization theorems are obtained.     

An improved Tau method for a class of Sturm-Liouville problems

We consider the numerical solution of Sturm-Liouville eigenvalue problems by Legendre-Gauss Tau method. The latter approximates the solution of differential equations as a finite sum of Legendre polynomials . We propose an improved approach which seeks approximants in terms of a finite sum of exponentially weighted Legendre polynomials for some real or complex frequencies {ω_k}. With the introduction of such exponentials, Legendre-Gauss Tau method can detect the sharp variations exhibited by the highly indexed Sturm-Liouville eigenfunctions. The efficiency of our results is illustrated through numerical examples.     

Left-definite boundary conditions of Sturm-Liouville problems when the interval shrinks to a point

On the condition that the interval of the problem shrinks to a point, we investigated the separated boundary conditions S α,β of left-definite Sturm-Liouville problem, and answered the following question: Is there a c 0 ∈ J such that S α,β is always left-definite or semi-left-definite for the Sturm-Liouville equation for each c ∈ (a, c 0 )?     

Nevanlinna theory of meromorphic functions on annuli

In this survey paper, we discuss the recent development of Nevanlinna theory of meromorphic functions on annuli, which extends results in Nevanlinna theory in the complex plane or in a disk. In particular, we show that the approach taken on annuli is a unified treatment of functions meromorphic in the complex plane, a disk and an annulus. It allows one to obtain many results in the complex plane and in a disk as corollaries of our results in annuli.     

On sampling theory associated with the resolvents of singular Sturm-Liouville problems

This paper is concerned with the sampling theory associated with resolvents of eigenvalue problems. We introduce sampling representations for integral transforms whose kernels are Green's functions of singular Sturm-Liouville problems provided that the singular points are in the limit-circle situation, extending the results obtained in the regular problems.

     

The best approximation to a class of functions of several variables by another class and related extremum problems

We study the relationship between several extremum problems for unbounded linear operators of convolution type in the spaces , m ≥ 1, 1 ≤ γ ≤ ∞. For the problem of calculating the modulus of continuity of the convolution operatorA on the function classQ defined by a similar operator and for the Stechkin problem on the best approximation of the operatorA on the classQ by bounded linear operators, we construct dual problems in dual spaces, which are the problems on, respectively, the best and the worst approximation to a class of functions by another class. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 323–340, September, 1998. This research was supported by INTAS under grant No. 94-4070.     

Products of generalized Nevanlinna functions with symmetric rational functions

New classes of generalized Nevanlinna functions, which under multiplication with an arbitrary fixed symmetric rational function remain generalized Nevanlinna functions, are introduced. Characterizations for these classes of functions are established by connecting the canonical factorizations of the product function and the original generalized Nevanlinna function in a constructive manner. Also, a detailed functional analytic treatment of these classes of functions is carried out by investigating the connection between the realizations of the product function and the original function. The operator theoretic treatment of these realizations is based on the notions of rigged spaces, boundary triplets, and associated Weyl functions.     

半平面中调和函数的积分表示

本文对于半平面中的调和函数u(z),证明了正部u (z)满足某些限制增长条件,用半平面边界上的积分表示,它的负部u-(z)也被类似的增长条件所控制.得到了半平面中负的调和函数的经典结果.     

A characterization of generalized poles of generalized Nevanlinna functions

Generalized poles of a generalized Nevanlinna function Q ∈ ??_κ (??) are defined in terms of the operator representation of Q . In this paper those generalized poles that are not of positive type and their degrees of non‐positivity are characterized analytically by means of pole cancellation functions. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic behavior for differences of eigenvalues of two Sturm-Liouville problems with smooth potentials

The asymptotics for the differences of the eigenvalues of two Sturm-Liouville problems defined on [0,π] with the same boundary conditions and different smooth potentials is considered. Under the assumptions of that both problems with a suite of boundary conditions have the same one full spectrum and both potential functions and their derivatives are the same at the endpoint x=π, the asymptotic expressions associated with other boundary conditions are provided.     

A generalized approximation theory for quadratic forms: Application to randomized spline type Sturm-Liouville problems

An approximation theory for families of quadratic forms is given. We show that if continuity conditions for a family of quadratic forms hold uniformly on an index set for the family, generalized signature approximation results hold. We then apply these results to randomized spline type Sturm-Liouville problems and obtain continuity of thenth eigenvalue for generalized Sturm-Liouville problems under weak hypotheses.     

Geometric properties of the self-adjoint Sturm-Liouville problems

In this paper, geometric properties of the self-adjoint Sturm-Liouville problems are investigated. It is proved that for linear self-adjoint Sturm-Liouville problems, the eigenfunctions correspond exactly to the projections of the curvature lines on the energy functional surface with an appropriate metric and that the eigenvalues correspond exactly to the principal curvatures (at the origin) of the same energy functional surface.     

A Schur transformation for functions in a general class of domains

In this paper we present a framework in which the Schur transformation and the basic interpolation problem for generalized Schur functions, generalized Nevanlinna functions and the like can be studied in a unified way. The basic object is a general class of functions for which a certain kernel has a finite number of negative squares. The results are based on and generalize those in previous papers of the first three authors on the Schur transformation in an indefinite setting.     

Nontrivial solutions of singular sublinear Sturm-Liouville problems

The singular sublinear Sturm-Liouville problems