Related papers: Integral representation of linear functionals on f…
By the Riesz representation theorem using the Riemann-Stieltjes integral, linear continuous functionals on the set of continuous functions from the unit interval into the reals can either be characterized by functions of bounded variation…
In this paper we study the integral representation in the space SBD(O) of special functions with bounded deformation of some L^1-norm lower semicontinuous functionals invariant with respect to rigid motions.
Let $\sigma_i$, $i=1,\ldots,n$, denote positive Borel measures on $\mathbb{R}^d$, let $\mathcal{D}$ denote the usual collection of dyadic cubes in $\mathbb{R}^d$ and let $K:\,\mathcal{D}\to[0,\infty)$ be a~map. In this paper we give…
Let $A(x): =(A_{i, j}(x))$ be a continuous function defined on some subshift of $\Omega:= \{0,1, \cdots, m-1\}^\mathbb{N}$, taking $d\times d$ non-negative matrices as values and let $\nu$ be an ergodic $\sigma$-invariant measure on the…
In [22], it was proved that as long as the integrand has certain properties, the corresponding It\^o integral can be written as a (parameterized) Lebesgue integral (or a Bochner integral). In this paper, we show that such a question can be…
In this work, we establish a representation theorem for multivariable totally symmetric functions: a multisymmetric continuous function must be the composition of a continuous function and a set of generators of the multisymmetric…
We give a complete characterization of all real-valued functions on the unit circle $S^1$ that can be represented by integrating the spherical distance on $S^1$ with respect to a signed measure or a probability measure.
The A-hypergeometric system studied by I.M. Gelfand, M.I. Graev, A.V. Zelevinsky and the author, is defined for a set A of characters of an algebraic torus. In this paper we propose a generalization of the theory where the torus is replaced…
It is explained how a locally convex (lc) topology $\tau$ on a real vector space $V$ extends to a locally multiplicatively convex (lmc) topology $\overline{\tau}$ on the symmetric algebra $S(V)$. This allows the application of the results…
In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continous convex functions on a vector space ${\bf R}^m$ and vector-valued functions in a weakly compact subset of a…
Let $\Omega\Subset\mathbb{C}^{n}$ be a domain with smooth boundary, $k\in\mathbb{N}$. It is shown that the integral of a holomorphic function in $L^1(\Omega)$ may be represented as the integral of this function against a smooth function…
Linear algebra's main concerns are sets of vectors, linear functions, subspaces, linear systems, matrices and concepts about those, such as whether the solution of linear system exists or is unique; a set of vectors is linearly independent…
The main purpose of this paper is to investigate some natural problems regarding the order structure of representable functionals on $^*$-algebras. We describe the extreme points of order intervals, and give a nontrivial sufficient…
We prove that a singular part $\mu_s$ of a measure $\mu$ satisfying ${\cal A}\mu =0$ for a linear partial differential operator ${\cal A}$ defined on $R^d$ has the range in the intersection of kernels of the principal symbol of ${\cal A}$…
This paper presents a point-free version of the Lebesgue integral for simple functions on $\sigma$-locales. It describes the integral with respect to a measure defined on the coframe of all $\sigma$-sublocales, moving beyond the constraints…
In this article, we characterize both Lusin's theorem and the existence of Borel representatives via the regularity properties of the measure in general topological measure spaces. As a corollary, we prove that Borel regularity of the…
Recent work on the representation of functions on sets has considered the use of summation in a latent space to enforce permutation invariance. In particular, it has been conjectured that the dimension of this latent space may remain fixed…
Let $(X, \mathscr{L}, \lambda)$ and $(Y, \mathscr{M}, \mu)$ be finite measure spaces for which there exist $A \in \mathscr{L}$ and $B \in \mathscr{M}$ with $0 < \lambda(A) < \lambda(X)$ and $0 < \mu(B) < \mu(Y)$, and let $I\subseteq…
This paper has two goals: to present some new results that are necessary for further study and applications of quasi-linear functionals, and, by combining known and new results, to serve as a convenient single source for anyone interested…
We introduce the notion of a gauge and of a tagged partition (subordinate to a given gauge) by intersections of open and closed sets of a compact metric space extending the corresponding notions in Henstock-Kurzweil integration of…