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In these lectures we are concerned with regularity theorems for functional equations and with some specific examples. For simplicity we consider functions defined in the real p-dimensional Euclidean space Rp with values in a Banach space X although most of the results are known in a more general setting.
Nous allons considérer des équations concernant des distributions inconnues correspondant aux équations fonctionnelles et cherchons à établir des méthodes pour trouver les solutions les plus générales (dans le domaine des distributions). Les équations que nous allons envisager se réduisent aux équations fonctionnelles usuelles sous la condition que les inconnues sont des fonctions (définissant des...
1. The present exposition is devoted to the motivations of a quite general system of functional equations in the foundations of Information Theory without probability. We shall deal with the concepts of uncertainty, expected information (entropy), conditional information. We define the measures of these quantities in such a manner that Information Theory can be considered quite independent from Probability...
En 1924 H. Prüfer [1] a introduit la notion de famille (Schar) comme une opération ternaire interne $$\begin{array}{*{20}c}{{\text{t}}\,{\text{ = }}\,{\text{f}}\left( {{\text{x,}}\,{\text{y,}}\,{\text{z}}} \right)\,,} \hfill & {} \hfill & {{\text{, y, z, t}} \in {\text{B}}} \hfill \\\end{array}$$ satisfaisant aux conditions suivantes I f(x,y,z) = x II f(x,y,z) = f(z,y,x) III...
1. In this paper I shall discuss the functional equation of meanvalue theory. For more details 1 refer to Chapter 15 of a forth coming treatise “Methods of Classical and Functional Analysis”, Ad dison-Wesley, 1971.
The main object of this paper is to develop a mathematical scheme, giving a formalization of a situation occurring in many interesting cases: some class of events is being observed by a set of observers, each observer evaluating the amount of information contained in one event according to his own scale. In every day life we can find examples of this situation; for instance, reading newspapers, we...
One of the most famous and important functional inequalities is the following one: 1 $${\text{f}}\left( {\frac{{{\text{x}}\,{\text{ + }}\,{\text{y}}}}{2}} \right) \leqslant \frac{{{\text{f}}\left( {\text{x}} \right)\,+ \,{\text{f}}\left( {\text{y}} \right)}}{2}$$ This inequality was first considered by Jensen [l5] and he gave to functions fulfilling (1) the name convex. Jensen himself expressed...
1. Let R be the additive group of all real numbers and t → Tt a representation of this group on an n-dimensional complex vector space X. This means that to each real number t a linear operator Tt : X → X is assigned in such a way that 1 $${\text{T}}_{{\text{t + s}}} = {\text{T}}_{\text{t}} {\text{T}}_{\text{s}} , {\text{T}}_0 = {\text{I}}$$ Since the set {Tt : t ∈ R} consists of operators which...
Introduction, Inequalities and functional equations are frequently used in probability theory. In the first section of this paper we discuss some important inequalities, in the second section we deal with certain functional equations encountered in probabilistic problems. The use of these tools is illustrated in section 3, where we derive a new result, namely a stability theorem for a characterization...
In these notes we consider problems of the following forms: let x, y, xi, yi, etc. denote elements of ℝn (we sometimes replace ℝn by a linear space χ), let t, r, ti., ri., etc. denote elements of ℝ Let f : ℝn →ℝ (occasionally, f : χ→y for anOtner linear space y ). Then our problems are of the $${\text{Type}}\,{\text{I}}:\left\{ {\begin{array}{*{20}c} {{\text{f(x)}}\,{\text{ = }}\,{\text{F}}\left\{...
1. Introduction: equations and inequalities. We are grateful to the organizers of this conference, both in a general, global, and in a special, local sense. Among the many thoughtful details I would like to single out Professor Forte's admonishment to start our courses with things understandable. This is best ensured by-telling what most of you know in a way that will naturally lead...
Let f(x) be monotonically decreasing for x ≥ 0, and non-negative. Denote by A the area bounded by the x-axis (0 ≤ x ≤ a), the positive y-axis (0 ≤ y ≤ f(0) ), and the graph of the function. The y-coordinate of the centre of gravity of A is given by 1 $$\eta = \frac{1}{2}\frac{{\int_0^{\text{a}} {\text{f}} \left( {\text{x}} \right)^2 {\text{dx}}}}{{\int_0^{\text{a}} {\text{f}} \left( {\text{x}} \right){\text{dx}}}}$$...
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