In this work we introduce a formulation for a non-local Hessian that combines the ideas of higher-order and non-local regularization for image restoration, extending the idea of non-local gradients to higher-order derivatives. By carefully choosing the weights, the model allows to improve on the current state of the art higher-order method, Total Generalized Variation, with respect to overall quality and particularly close to jumps in the data. In the spirit of recent work by Brezis et al., our formulation also has analytic implications: for a suitable choice of weights, it can be shown to converge to classical second-order regularizers, and in fact allows a novel characterization of higher-order Sobolev and BV spaces.

For the classical space of functions with bounded mean oscillation, it is well known that $\operatorname{VMO}^{**} = \operatorname{BMO}$ and there are many characterizations of the distance from a function$f$in $\operatorname{BMO}$ to $\operatorname{VMO}$ . When considering the Bloch space, results in the same vein are available with respect to the little Bloch space. In this paper such duality results and distance formulas are obtained by pure functional analysis. Applications include general Möbius invariant spaces such as$Q$_{$K$}-spaces, weighted spaces, Lipschitz–Hölder spaces and rectangular $\operatorname{BMO}$ of several variables.

In this paper we continue to advance the theory regarding the Riesz fractional gradient in the calculus of variations and fractional partial differential equations begun in an earlier work of the same name. In particular we here establish an L1 Hardy inequality, obtain further regularity results for solutions of certain fractional PDE, demonstrate the existence of minimizers for integral functionals of the fractional gradient with non-linear dependence in the field, and also establish the existence of solutions to corresponding Euler-Lagrange equations obtained as conditions of minimality. In addition we pose a number of open problems, the answers to which would fill in some gaps in the theory as well as to establish connections with more classical areas of study, including interpolation and the theory of Dirichlet forms.

This paper deals with the moment problem on a (not necessarily finitely generated) commutative unital real algebra A. We define moment functionals on A as linear functionals which can be written as integrals over characters of A with respect to cylinder measures. Our main results provide such integral representations for A+–positive linear functionals (generalized Haviland theorem) and for positive functionals fulfilling Carleman conditions. As an application, we solve the moment problem for the symmetric algebra S(V) of a real vector space V. As a byproduct, we obtain new approaches to the moment problem on S(V) for a nuclear space V and to the integral decomposition of continuous positive functionals on a barrelled nuclear topological algebra A.

To study arithmetic structures of natural numbers, we introduce a notion of entropy of arithmetic functions, called anqie entropy. This entropy possesses some crucial properties common to both Shannon's and Kolmogorov's entropies. We show that all arithmetic functions with zero anqie entropy form a C*-algebra. Its maximal ideal space defines our arithmetic compactification of natural numbers, which is totally disconnected but not extremely disconnected. We also compute the $K$-groups of the space of all continuous functions on the arithmetic compactification. As an application, we show that any topological dynamical system with topological entropy $\lambda$, can be approximated by symbolic dynamical systems with entropy less than or equal to $\lambda$.