Operator Functional Equations in Analysis

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 Ck-spaces or Schwartz spacesℝn. Naturally, the results strongly depend on the type of the domain and the image space. © Springer Science+Business Media New York 2013.