On singly-periodic minimal surfaces with planar ends

The spaces of nondegenerate properly embedded minimal surfaces in quotients of by nontrivial translations or by screw motions with nontrivial rotational part, fixed finite topology and planar type ends, are endowed with natural structures of finite dimensional real analytic manifolds. This nondegeneracy is defined in terms of Jacobi functions. Riemann's minimal examples are characterized as the only nondegenerate surfaces with genus one in their corresponding spaces. We also give natural immersions of these spaces into certain complex Euclidean spaces which turn out to be Lagrangian immersions with respect to the standard symplectic structures.

     

Generalized Convolution for the Discrete Jacobi Transformation

We study the translation and the convolution associated to the discrete Jacobi transformation on complex sequences of slow and rapid growth. Also we establish new topological properties for these spaces of sequences. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotics of Eigenvalues for a Class of Jacobi Matrices with Limiting Point Spectrum

In this paper, we study a class of Jacobi matrices with very rapidly decreasing weights. It is shown that the Weyl function (the matrix element of the resolvent of the operator) for the class under study can be expressed as the ratio of two entire transcendental functions of order zero. It is shown that the coefficients in the expansion of these functions in Taylor series are proportional to the generating functions of the number of integral solutions defined by certain Diophantine equations. An asymptotic estimate for the eigenvalues is obtained.     

Self-adjointness of unbounded tridiagonal operators and spectra of their finite truncations

This paper addresses two different but related questions regarding an unbounded symmetric tridiagonal operator: its self-adjointness and the approximation of its spectrum by the eigenvalues of its finite truncations. The sufficient conditions given in both cases improve and generalize previously known results. It turns out that, not only self-adjointness helps to study limit points of eigenvalues of truncated operators, but the analysis of such limit points is a key help to prove self-adjointness. Several examples show the advantages of these new results compared with previous ones. Besides, an application to the theory of continued fractions is pointed out.     

The higher-order differential operator for the generalized Jacobi polynomials — new representation and symmetry

For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by Koornwinder’s generalized Jacobi polynomials with four parameters $\alpha ,\beta \in {\mathbb{N}}_0$ and $M,N\ge 0$ determining the orthogonality measure on the interval $?1\le x\le 1$. The corresponding differential equation of order $2\alpha +2\beta +6$ is presented here as a linear combination of four elementary components which make the corresponding differential operator widely accessible for applications. In particular, we show that this operator is symmetric with respect to the underlying scalar product and thus verify the orthogonality of the eigenfunctions.     

Uncertainty principles for Jacobi expansions

In this paper an uncertainty principle for Jacobi expansions is derived, as a generalization of that for ultraspherical expansions by Rösler and Voit. Indeed a stronger inequality is proved, which is new even for Fourier cosine or ultraspherical expansions. A complex base of exponential type on the torus related to Jacobi polynomials is introduced, which are the eigenfunctions both of certain differential-difference operators of the first order and the second order. An uncertainty principle related to such exponential base is also proved.     

Convolution structure associated with the Jacobi-Dunkl operator on IR

In this paper, a product formula for the eigenfunction of the Jacobi-Dunkl differential-difference operator is derived. It leads to a uniformly bounded convolution of point measures and a signed hypergroup on IR. 2000 Mathematics Subject Classification Primary—34K99, 44A15, 44A35, 43A15     

Direct and inverse multichannel scattering problems

We consider small oscillations of a system of pairwise interacting particles in an external field near a stable equilibrium. The system is assumed to consist of finitely many channels, i.e., semi-infinite linear chains of particles, attached to a scatterer, which is a finite system of interacting particles. Direct and inverse scattering problems are considered. In particular, an algorithm finding characteristics of the channels on the basis of scattering data is given.     

Nonsymmetric Osserman pseudo-Riemannian manifolds

Examples of Osserman pseudo-Riemannian manifolds with metric of any signature , , which are not locally symmetric are exhibited.

     

Polynomial approximation of functions with given order of thekth generalized modulus of smoothness

The problem of approximation by algebraic polynomials is considered on function classes characterized by the value of thekth generalized modulus of smoothness defined by the Jacobi generalized shift operator. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 425–436, March, 1998.     

SINGULAR INTEGRAL OPERATORS IN L~2 SPACES ASSOCIATED WITH THE GENERALIZED JACOBI WEIGHTS

In this paper, a kind of new definitions of singular integral operators in the weighted L2 space with the generalized Jacobi weights are given, then the boundedness in the weighted L2 space, the relationships between these operators and the classic singular integral operators are proved. For the convenience in the later use, similar results in the L2 space with Jacobi weights are given.     

Spectra of infinite graphs with tails

We compute explicitly (modulo solutions of certain algebraic equations) the spectra of infinite graphs obtained by attaching one or several infinite paths to some vertices of given finite graphs. The main result concerns a canonical form for the adjacency matrix of such infinite graphs, and the algorithm of its calculation. The argument relies upon the spectral theory of eventually free Jacobi matrices. We also study some other couplings of infinite graphs (stars and Bethe–Caley trees).     

Real hypersurfaces in complex projective space with recurrent structure Jacobi operator

We classify real hypersurfaces of complex projective space , m3, with -recurrent structure Jacobi operator and apply this result to prove the non-existence of such hypersurfaces with recurrent structure Jacobi operator.     

Some Continuous Analogs of the Expansion in Jacobi Polynomials and Vector-Valued Orthogonal Bases

We obtain the spectral decomposition of the hypergeometric differential operator on the contour Re z = 1/2. (The multiplicity of the spectrum of this operator is 2.) As a result, we obtain a new integral transform different from the Jacobi (or Olevskii) transform. We also construct an ₃ F ₂-orthogonal basis in a space of functions ranging in ℂ². The basis lies in the analytic continuation of continuous dual Hahn polynomials with respect to the index n of a polynomial.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 31–46, 2005Original Russian Text Copyright © by Yu. A. Neretin     

A characterization of Clifford tori with constant scalar curvature one by the first stability eigenvalue

Let M be a compact hypersurface with constant scalar curvature one immersed into the unit Euclidean sphere . As is well-known, such hypersurfaces can be characterized variationally as critical points of the integral _M Hdv. In this paper we derive a sharp upper bound for the first eigenvalue of the corresponding Jacobi operator in terms of the mean curvature of the hypersurface. Moreover, we prove that this bound is achieved only for the Clifford tori in with scalar curvature one.—Dedicated to the memory of Prof. José F. Escobar, Chepe     

复射影空间和四元数射影空间中广义赤道的一个新特征

本文研究复射影空间ＣＰ＾ｎ和四元数射影空间ＱＰ＾ｎ中的紧致极小超曲面，对其上Ｊａｃｏｂｉ算子的最小特征值给出了一个最佳上界估计，由此给出了ＣＰ＾ｎ和ＱＰ＾ｎ中广义赤道的一个新的特征。     

REPRESENTATION OF GENERALIZED TRANSLATIONS OF THE PRODUCT OF TWO FUNCTIONS ON SIGNED HYPERGROUPS ON THE REAL LINE

In this paper,we consider the generalized translations associated with the Dunkl and the Jacobi-Dunkl differential-difference operators on the real line which provide the structure of signed hrpergroups on R.Especially,we study the representation of the generalized translations of the product of two functions for these signed hypergroups.     

On conformal biharmonic immersions

This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a two-parameter family of non-minimal conformal biharmonic immersions of cylinder into and some examples of conformal biharmonic immersions of four-dimensional Euclidean space into sphere and hyperbolic space, thus providing many simple examples of proper biharmonic maps with rich geometric meanings. These suggest that there are abundant proper biharmonic maps in the family of conformal immersions. We also explore the relationship between biharmonicity and holomorphicity of conformal immersions of surfaces.