The goal of this paper is two-fold. First we study theoretical properties of discrete spherical means of directional derivatives of a function. Then we focus on the two-dimensional case and use discrete circular means to derive rotation-equivariant discrete approximations of linear and nonlinear firstand second-order differential operators. Applications to nonlinear filtering of digital images ...

We consider multifunctions acting between two linear normed spaces and having closed convex images. Approximations are considered which serve as an expansion of it. Generalized delta theorems for random sets in in nite dimensions are shown using those approximations. Furthermore, univalued, resp. Castaing representations of the multifunction are constructed with (higher order) di erentiability ...

In this note we give a subdifferential mean value inequality for every continuous Gâteaux subdifferentiable function f in a Banach space which only requires a bound for one but not necessarily all of the subgradients of f at every point of its domain. We also give a subdifferential approximate Rolle’s theorem stating that if a subdifferentiable function oscillates between −ε and ε on the bounda...

Two properties concerning the space of differences of sublinear functions D(X) for a real Banach space X are proved. First, we show that for a real separable Banach space (X, ‖ · ‖) there exists a countable family of seminorms such that D(X) becomes a Fréchet space. For X = R this construction yields a norm such that D(R) becomes a Banach space. Furthermore, we show that for a real Banach space...

Applications of Fr?chet derivative emerge in the sensitivity analysis matrix functions. Our work extends generalized complex step approximation using computation f (A + ei?hE) as a tool to case, and combines it with finite difference formula estimate derivative. We provide numerical results for first second order exponential square root.

K e y w o r d s N e u t r a l functional differential equation, Analytic semigroup, Mild solution, FaedoGalerkin approximation. 1. I N T R O D U C T I O N Let H be a separable Hilbert space and 0 < T, T < ec. For 0 < t < T, Ct := C([-~-, t]; H) be the Banach space of all continuous functions from l--T, t] into H endowed with the supremum norm, Ilullt := sup I lu(s) l l , u • Ct, (1.1) -r<_s<t w...

In this paper, we investigate minimal (weak) approximate Hessians, and completely answer the open questions raised by V. Jeyakumar and X. Q. Yang. As applications, we first give a generalised Taylor’s expansion in terms of a minimal weak approximate Hessian. Then we characterise the convexity of a continuously Gâteaux differentiable function. Finally some necessary and sufficient optimality con...