Interpolation by Analytic Matrix Functions

There are some classical interpolation problems for complex functions, solved many times over decades ago, which still generate dozens of new papers. One reason for this is strictly mathematical; the development of certain branches of operator theory has enabled us to view the problems in a different way and so understand some aspects of them better. There is another reason, which for me rings an even louder bell: contact with the theory of circuits and systems developed by engineers. An enormous range of worthwhile problems about complex functions comes from this source. Many of them are close to questions studied by the old masters of analysis: for example, variants of the Nevanlinna-Pick interpolation problem arise from a remarkable diversity of starting points. The earliest instance I know dates back to 1940 (see the account of F. Fenyves’ work in [9]) while, more recently, J.W. Helton’s far-reaching application of non-Euclidean functional analysis to electronics also centres around this problem [4]. However, the engineering slant generally calls for something slightly different from the old results.