Second Order Polynomials in the Two Dimensional Moment Problem

Abstract

Background. The properties of block Jacobi matrices corresponding two-dimensional moment problem are studied here. We introduce polynomials of the second kind similar to the corresponding polynomials of the second kind corresponding classical Hamburger moment problem. In a previous publication we orthogonalize two-index family polynomial \[x^{n}, y^{n}, n,m\in \textrm{N}_{0}\] with respect to the measure on the real plane. The resulting polynomials \[P_{n,\alpha }(x,y),\alpha =0, 1, ...,n\] are the analogues polynomials of the first kind. The same polynomials are solutions of the system of difference equations \[J_{A} P(x,y)=xP(x,y),J_{B} P(x,y)=yP(x,y)\] generated by symmetric block Jacobi matrices \[J_{A}\] and \[J_{B},\] corresponding operators are commute in the strong resolvent sense. Solutions exist for a given initial condition, i.e., first polynomial is supposed constant for certainty equal unit \[P_{0;0}(x,y)=1.\] Our investigations consist of the fact to confirm or refute the hypothesis that the second kind polynomial \[Q_{n;\alpha}(x,y)\] is also satisfy the same system but with another initial condition – first polynomial is constant equal zero \[Q_{0;0}(x,y)=0.\] Polynomials of the second kind in the classical case are defined by a certain functional.

Objective. The purpose of the study is to find functional that would define polynomials of the second kind, using given by polynomials of the first kind. Thus obtained polynomials of the second kind must also satisfy the system of difference equations.

Methods. Getting results are contributed to numerous examples of consideration, partial eases. Next verified.

Results. The result of research is suggested in this functional analogue of the two-dimensional case:

\[Q_{n} (z_{1},z_{2})=\iint_{R^{2}}\frac{P_{n}(\lambda ,\mu )-P_{n}(\lambda,z_{2})-P_{n}(z_{1},\mu)+P_{n}(z_{1},z_{2})}{(\lambda -z_{1})(\mu -z_{2})} d\rho (\lambda,\mu )\]

where \[Q_{n} (z_{1},z_{2})=(Q_{n;0} (z_{1},z_{2}), Q_{n;1} (z_{1},z_{2}),...,Q_{n;n} (z_{1},z_{2})),z_{1},z_{2}\in C\setminus R, n\in N_{0}.\]

Conclusions. This paper introduced polynomials of the second kind related to real two-dimensional moment problem. It is shown that these polynomials satisfy a system of difference equations generated by block Jacobi matrices type. For polynomials of the first kind the convergence of the series is studied based on the certainty or uncertainty investigated problem points.