Rational interpolation and mixed inverse spectral problem for finite CMV matrices

Abstract

For finite-dimensional CMV matrices the mixed inverse spectral problem of reconstructing the matrix by its submatrix and a part of its spectrum is considered. A general rational interpolation problem which arises in solving the mixed inverse spectral problem is studied, and the description of the space of its solutions is given. We apply the developed technique to give sufficient conditions for the uniqueness of the solution of the mixed inverse spectral problem.

An inverse spectral theory for finite CMV matrices

“…Finally, in Section 4 we solve the inverse problem of reconstructing the CMV matrix by the two spectra, examined in the preceding section. In our forthcoming paper [12] we consider the mixed inverse problems for finite CMV matrices.…”

Inverse spectral analysis for a class of infinite band symmetric matrices

“…Although the inverse spectral problems for Jacobi matrices have been studied extensively (see for instance [7-9, 14, 19, 22-24, 34-36] for the finite case and [10,11,13,14,20,21,37,38] for the infinite case), works dealing with band matrices non-necessary tridiagonal are not so abundant (see [5,17,18,[27][28][29]32,40,41] for the finite case and [3,16] for the infinite case).…”

“…In this work, the approach to the inverse spectral analysis of the operators whose matrix representation belongs to M(n, N) is based on the one used in [24,25], but it allows to treat the case of arbitrary n. An important ingredient of the method used here is the linear interpolation of n-dimensional vector poly-Figure 3: Mass-spring system of a matrix in M(2, 10): nondegenerated case Figure 4: Mass-spring system of a matrix in M(2, 10): degenerated case nomials, recently developed in [26]. The linear interpolation theory of [26] is a nontrivial generalization of the rational interpolation theory developed in [15] from ideas given in [24,25]. It is on the basis of the results of [26] that the inverse spectral theory developed in [24,25] is extended here to band matrices with 2n + 1 diagonals (n ∈ N).…”

On a linear interpolation problem for n-dimensional vector polynomials

“…The results of this paper give a complete characterization of all solutions of the interpolation problem (1). The interpolation problem defined above has been studied in [7] and much earlier in [11] for the particular setting when n = 2. In this case, the theory developed in [7,11] allows to treat the problem of finding a rational function P 1 (z)/P 2 (z) which takes the value −α 2 (j)/α 1 (j) ∈ C at each interpolation node z j .…”

“…Similarly, the spectral analysis of five-diagonal symmetric matrices also demands additional conditions (see [8]) on the rational interpolation problem. The description obtained in [7] permits to reduce the rational interpolation problem with such additional restrictions to a triangular linear system and to answer the specific question of the existence and uniqueness (or non-uniqueness) of the solution to the inverse spectral problem. Other approaches to rational interpolation can be found in [4,5,10].…”