Matrix Nevanlinna-Pick-Takagi Interpolation

Abstract

We have seen in the previous chapter that, in order that there exist a (scalar) rational function on the unit disk D with modulus less than 1 there which in addition satisfies some interpolation conditions

$$f({z_i}) = {\omega _i},\;1 \leqslant i \leqslant n,$$

(1)

it is necessary and sufficient that the associated Pick matrix

$$\Lambda = {\left[{\frac{{1 - {{\bar w}_i}{w_j}}}{{1 - {{\bar z}_i}{z_j}}}} \right]_{1 \leqslant i,j \leqslant n}}$$

be positive definite Thus if Λ has some negative eigenvalues, such a function f cannot exist. A natural question is what can still be said about the class of interpolating functions if Λ has some negative eigenvalues. The solution turns out to be rather elegant. Namely, if к is the number of negative eigenvalues of Λ, then one can always find a rational function f with к poles in D and with modulus on the boundary ∂D at most 1 which satisfies the interpolation conditions (1) at each z _i which is a point of analyticity. Conversely, Λ has к negative eigenvalues whenever such an f exists. This type of result was first obtained by Takagi [1924] in the context of the Schur problem where one specifies the first few Taylor coefficients of f at the origin.