Operator Functional Equations in Analysis

Chapter
Part of the Fields Institute Communications book series (FIC, volume 68)

Abstract

Classical operations in analysis and geometry as derivatives, the Fourier transform, the Legendre transform, multiplicative maps or duality of convex bodies may be characterized, essentially, by very simple properties which may be often expressed as operator equations, like the Leibniz or the chain rule, bijective maps exchanging products with convolutions or bijective order reversing maps on convex functions or convex bodies. We survey and discuss recent results of this type in analysis. The operations we consider act on classical spaces like C k -spaces or Schwartz spaces \(\mathcal{S}({\mathbb{R}}^{n})\). Naturally, the results strongly depend on the type of the domain and the image space.

Key words

Operator equations Leibniz rule Chain rule Multiplicative maps Fourier transforms 

Notes

Acknowledgements

Hermann König was partially supported by the Fields Institute and Vitali Milman was partially supported by the Minkowski Center at the University of Tel Aviv, by the Fields Institute, by ISF grant 387/09 and BSF grant 2006079.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Mathematisches SeminarUniversität KielKielGermany
  2. 2.Department of Mathematics, School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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