Vol. 27, No. 2, 1968

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On matrices associated with generalized interpolation problems

Douglas Napier Clark

Vol. 27 (1968), No. 2, 241–253

Two classical interpolation theorems, due to Carathéodory-Fejér and Nevanlinna-Pick, deal with classes of functions analytic in th unit disk which take certain prescribed values at finitely many points there. The theorems express certain extrema of these classes as eigenvalues of finite matrices. In this paper, there is given a generalization of this type of interpolation, which involves inner functions. It is seen that a certain theorem about Hankel matrices and projections of Toeplitz matrices generalizes both of the above interpolation theorems. The theorem also provides a generalization of some recent work of the author on meromorphic interpolation and a continuous analogue of a theorem on Toeplitz forms and interpolation. Finally, the theorem has some consequences in the theory of infinite Hankel matrices.

Mathematical Subject Classification
Primary: 30.67
Received: 11 December 1967
Published: 1 November 1968
Douglas Napier Clark