Interpolation with positive real functions of several variables

Abstract

AbstractThe interpolation problem for n‐dimensional positive real functions is formulated in two different settings: with interpolation knots either contained on the distinguished boundary of the Cartesian product of open right half planes or in this n‐dimensional ‘open right half plane’. A complete solution of the first problem is given. Necessary conditions of existence of a solution of the second problem are proved and the well known Nevanlinna‐Pick algorithm is generalized so that construction of interpolating n‐dimensional positive real functions becomes possible at least in case when the interpolation knots are real vectors. Examples illustrate the suggested methods, show the complexity and significance of the problem and methods under discussion.Interpolation in special classes of functions has for some time been an important problem in mathematics. General schemes of solution in linear functional spaces with orthogonality are well known. For classes of functions which do not satisfy the axioms of a linear space special methods have been studied in detail, because they are closely connected to other important parts of mathematics such as approximation theory, numerical analysis and the theory of moments. Considerable efforts have been put into interpolation in the complex domain with functions analytic in the unit disc and with positive real functions. the significance of interpolation procedures in the class of one‐variable positive real functions in network theory, e.g. in broadband matching and approximation problems has been recognized in References 1‐3. an excellent survey paper has recently been devoted to these problems;4 in this paper some new ideas for the use of interpolation in system theory were outlined. the growing interest in multivariable network theory calls for generalizations of interpolation procedures to multivariable cases. Many of the one‐variable applications can easily be extended to the multivariable case and it is hoped that in passive network theory and in problems of n‐dimensional digital systems the solution of the interpolation problem will contribute to the difficult synthesis problems in the multivariable theory.In this paper conditions for the existence of multivariable positive real interpolating functions will be investigated and some procedures for the construction of such functions will be described.