On the approximation of harmonic functions by harmonic polynomials

Siberian Mathematical Journal -     

On one set of orthogonal harmonic polynomials

A new basis of harmonic polynomials is given. Proposed polynomials are orthogonal on the unit sphere and each term of this basis consists of monomials not present in the others.

     

On the number of zeros of certain harmonic polynomials

Using techinques of complex dynamics we prove the conjecture of Sheil-Small and Wilmshurst that the harmonic polynomial , 1$">, has at most complex zeros.

     

Kravchuk polynomials and group representations

Kravchuk polynomials of a discrete variable and different types of Kravchuk q-polynomials are determined. The importance of these polynomials for the theory of representations of Lie and Chevalley groups and the theory of symmetric and quantum groups is shown. The article presents a survey of the contemporary results in these areas.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 7, pp. 888–901, July, 1992.     

Product representations of polynomials

For a fixed polyomial , let ρ_k(N) denote the maximum size of a set A{1,2,…,N} such that no product of k distinct elements of A is in the value set of f. In this paper, we determine the asymptotic behaviour of ρ_k(N) for a wide class of polynomials. Our results generalize earlier theorems of Erdős, Sós and Sárközy.     

Explicit representations of biorthogonal polynomials

Given a parametrised weight function (x,) such that the quotients of its consecutive moments are Möbius maps, it is possible to express the underlying biorthogonal polynomials in a closed form [5]. In the present paper we address ourselves to two related issues. Firstly, we demonstrate that, subject to additional assumptions, every such obeys (inx) a linear differential equation whose solution is a generalized hypergeometric function. Secondly, using a generalization of standard divided differences, we present a new explicit representation of the underlying biorthogonal polynomials.     

Principal series representations and harmonic spinors

Let G be a real reductive Lie group and G/H a reductive homogeneous space. We consider Kostant's cubic Dirac operator D on G/H twisted with a finite-dimensional representation of H. Under the assumption that G and H have the same complex rank, we construct a nonzero intertwining operator from principal series representations of G into the kernel of D. The Langlands parameters of these principal series are described explicitly. In particular, we obtain an explicit integral formula for certain solutions of the cubic Dirac equation D=0 on G/H.     

Integral representations for the generalized Bedient polynomials and the generalized Cesàro polynomials

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing our main results, the corresponding integral representations are deduced for such familiar classes of hypergeometric polynomials as (for example) the generalized Bedient polynomials and the generalized Cesàro polynomials. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.     

The valence of harmonic polynomials

The paper gives an upper bound for the valence of harmonic polynomials. An example is given to show that this bound is sharp.

     

Several polynomials associated with the harmonic numbers

We develop polynomials in z∈C for which some generalized harmonic numbers are special cases at z=0. By using the Riordan array method, we explore interesting relationships between these polynomials, the generalized Stirling polynomials, the Bernoulli polynomials, the Cauchy polynomials and the Nörlund polynomials.     

Integral representations of analytic and harmonic functions

Doklady Mathematics -     

Singular unitary representations and indefinite harmonic theory

Square-integrable harmonic spaces are defined and studied in a homogeneous indefinite metric setting. In the process, Dolbeault cohomologies are unitarized, and singlar unitary representations are obtained and studied.     

Strongly inequivalent representations and Tutte polynomials of matroids

We develop constructive techniques to show that non-isomorphic 3-connected matroids that are representable over a fixed finite field and that have the same Tutte polynomial abound. In particular, for most prime powers q, we construct infinite families of sets of 3-connected matroids for which the matroids in a given set are non-isomorphic, are representable over GF(q), and have the same Tutte polynomial. Furthermore, the cardinalities of the sets of matroids in a given family grow exponentially as a function of rank, and there are many such families.In Memory of Gian-Carlo Rota