Result: Characteristics of spectra of higher order difference operators and the convergence of the joint rational approximations

The issue concerning the relations between joint Hermite-Pade approximations and polynomials determined by orthogonal relations systems on one hand, and operator spectral theory on the other hand, has been open for a long time. In this article, we present results obtained in that direction. It has been found that the convergence of joint approximations is connected with the characteristics of the spectrum of some difference operator of \(p+1\) order, which, in an appropriate basis, can be represented in the form of an essentially asymmetric \((p+2)\)-diagonal matrix. In our presentation of the results we constrain ourselves to the case \(p=2\), and separately note the modifications needed for the case \(p>2\).