Generic Differentiability of Lipschitzian Functions

Generic G&eaux- Differentiability of Convex Functions on Small Sets

The theory of generic differentiability of convex functions on Banach spaces is by now a well-explored part of infinite-dimensional geometry. All the attempts to solve this kind of problem have in common, as a working hypothesis, one special feature of the finite-dimensional case. Namely, convex functions are always considered to be defined on convex sets with nonempty interior. But typically, ...

Generic Fr6chet -differentiability of Convex Functions on Small Sets

On Fréchet Differentiability of Lipschitzian Functions on Spaces with Gaussian Measures

We construct two counter-examples related to Fréchet differentiability in infinite dimensions. The first one gives a convex Lipschitzian function on a Banach space such that its convolution with a given measure is Fréchet differentiable only on a measure zero set. The second one gives a Borel function on a space with a Gaussian measure such that it is Lipschitzian along the Cameron–Martin subsp...

Differentiability of Functions of MatricesMatrix Functions DIFFERENTIABILITY OF FUNCTIONS OF MATRICES

A. Let f be a function on the set of diagonal n × n matrices, and let˜f be the unique extension of f to the set of symmetric n × n matrices invariant with respect to conjugation by orthogonal matrices. We show that˜f has the same regularity properties as f.

Differentiability of Functions of Contractions

In this paper we study differentiability properties of the map T 7→ φ(T ), where φ is a given function in the disk-algebra and T ranges over the set of contractions on Hilbert space. We obtain sharp conditions (in terms of Besov spaces) for differentiability and existence of higher derivatives. We also find explicit formulae for directional derivatives (and higher derivatives) in terms of doubl...