Description For Nevanlinna-Pick Interpolation Problem By Linear Fractional Transformation

Reproducing kernel space has many good properties, in which regeneration is the most basic property of the reproducing kernel function. Meanwhile, according to difference in the selected reproducing kernel function, reproducing kernel space can be divided into many types, and H ( s ) and H (U )are reproducing kernel Hilbert spaces on two typical vector-valued functions. Although they have some different properties, there exixts close relation between these two reproducing kernel Hilbert spaces, namely, for any element in H (U ), there is a matrix-valued function and let it be left-multiplied by the element in H (U ), then we can obtain the new element which exactly belongs to H ( s ). This relation between these two spaces and the regeneration of reproducing kernel function play a key role in this article.There are lots of good properties for linear fractional transformation defined in terms of block matrix. It is not only a part of reproducing kernel Hilbert space theory, but also a powerful means to research reproducing kernel Hilbert space theory. Particularly, in Schur class S p×q(Ω), we can apply linear fractional transformation defined by J contraction matrix to describe the solution sets for many kinds of one sided tangential anf bitangential interpolation problem.The special classes of reproducing kernel Hilbert spaces H ( s )and H (U ) that were introduced and extensively studied by L.de Branges are mainly used in this paper. Firstly, the characterization of linear fractional transformation set T_U [ S ^p×q] is studied. Secondly, this paper deals with the sufficient and necessary condition of solvable classical Nevanlinna-Pick interpolation problem in the case of matrix in Schur class S ^p×q(Ω) by means of reproducing kernel space theory, and then in terms of linear fractional transformation, the characterization of solution set for Nevanlinna-Pick type interpolation problem is described separately on open upper semi-plane, open right semi-plane, and open unit disc. At last, when block J contraction matrix in linear fractional transformation is invertible, the another kind of characterization of solution set for Nevanlinna-Pick type interpolation problem is presented by elements in the inverse block matrix.