About this book
This book provides an in depth discussion of Loewner’s theorem on the characterization of matrix monotone functions. The author refers to the book as a ‘love poem,’ one that highlights a unique mix of algebra and analysis and touches on numerous methods and results. The book details many different topics from analysis, operator theory and algebra, such as divided differences, convexity, positive definiteness, integral representations of function classes, Pick interpolation, rational approximation, orthogonal polynomials, continued fractions, and more. Most applications of Loewner’s theorem involve the easy half of the theorem. A great number of interesting techniques in analysis are the bases for a proof of the hard half. Centered on one theorem, eleven proofs are discussed, both for the study of their own approach to the proof and as a starting point for discussing a variety of tools in analysis. Historical background and inclusion of pictures of some of the main figures who have developed the subject, adds another depth of perspective.
The presentation is suitable for detailed study, for quick review or reference to the various methods that are presented. The book is also suitable for independent study. The volume will be of interest to research mathematicians, physicists, and graduate students working in matrix theory and approximation, as well as to analysts and mathematical physicists.
matrix convex approximation theory Pick interpolation Cauchy interpolation Loewner's theorem monotone matrix function Herglotz representation Simon Grundlehren Lieb concavity Donoghue theorem
- DOI https://doi.org/10.1007/978-3-030-22422-6
- Copyright Information Springer Nature Switzerland AG 2019
- Publisher Name Springer, Cham
- eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
- Print ISBN 978-3-030-22421-9
- Online ISBN 978-3-030-22422-6
- Series Print ISSN 0072-7830
- Series Online ISSN 2196-9701
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