Abstract
The study of the exact interpolation of quadratic norms in vector spaces depends in an essential way on the theory of monotone matrix functions developed by Loewner in 1934 [4]. This theory, in its turn, depends on Loewner’s solution of a problem of interpolation by rational functions of a certain class. The discussion of this latter problem is necessarily complicated, and Loewner’s text does not lend itself to ready reference. It has therefore seemed worthwhile to recast a portion of Loewner’s results in a form more suited to the applications we have in view. Our work, however, is not wholly derivative; none of our theorems are explicitly stated by Loewner and our arguments, which are of a more geometric character, are essentially different. The knowledgeable reader will note that our hypotheses are slightly stronger than Loewner’s and that our results are therefore also stronger. For the applications which we have in mind, Theorem III is the most important result; the proof of this theorem depends on all of the previously developed theory.
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References
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Donoghue, W.F. The theorems of Loewner and Pick. Israel J. Math. 4, 153–170 (1966). https://doi.org/10.1007/BF02760074
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DOI: https://doi.org/10.1007/BF02760074