The present paper is a continuation of [5, 7] where a Fredholm theory for approximation sequences is proposed and some of its properties and consequences are studied. Here this theory is specified to the class of fractal approximation methods. The main result is a formula for the so-called α-number of an approximation sequence (Aₙ) which is the analogue of the kernel dimension of a Fredholm operator.

Nella prima parte della nota sono studiate le proprietà di connessione dei sottospazi dello spettro di un anello. Con l’ausilio dei risultati ottenuti, si stabiliscono le condizioni necessarie e sufficienti affinchè un’algebra reale assolutamente piatta sia isomorfa ad un’algebra di funzioni continue a valori reali su un $P$-spazio, del quale determini la topologia. Ulteriori condizioni sono necessarie e sufficienti affinché un’algebra reale assolutamente piatta sia isomorfa all’algebra di tutte le...

Soit $\mathbf{A}$ un ensemble quelconque d’opérateurs différentiels en deux variables à coefficients complexes constants. Soit $C_0$ l’espace des fonctions continues complexes tendant vers zéro à l’infini dans le plan euclidien. Soit $C_0\left(\mathbf{A}\right)$ l’espace $\{f:f\in C_0,A_f{C}_0$, tout $a\in \mathbf{A}\}$. Classifier ces espaces équivaut à trouver des conditions nécessaires et suffisantes sur des opérateurs différentiels $P_1,...,P_n$ pour que $\parallel P_1\phi {\parallel }_{\infty }\le K(\parallel P_2\phi {\parallel }_{\infty }+\cdots +\parallel P_n\phi {\parallel }_{\infty })$. Il paraît que ce problème général est bien difficile. Nous présentons ici la solution complète dans le cas spécial des $C_0\left(\mathbf{A}\right)$ stables...

La structure d’une variété $V$ indéfiniment différentiable est complètement caractérisée par l’algèbre des fonctions indéfiniment différentiables sur $V$. Pour des surfaces de Riemann il n’y a pas, en général, une algèbre caractérisante canonique de fonctions globalement définies. Dans ce travail l’on définit une classe dénombrable de telles algèbres. Ces algèbres sont des analogues, pour les surfaces de Riemann, des algèbres définies pour le plan par les auteurs dans “Algebras of differentiable functions...

A systematic investigation of algebras of holomorphic functions endowed with the Hadamard product is given. For example we show that the set of all non-invertible elements is dense and that each multiplicative functional is continuous, answering some questions in the literature.

We deal with the algebras consisting of the quotients that produce bounded evaluation on suitable ideals of the multiplication algebra of a normed semiprime algebra A. These algebras of quotients, which contain A, are subalgebras of the bounded algebras of quotients of A, and they have an algebra seminorm for which the relevant inclusions are continuous. We compute these algebras of quotients for some norm ideals on a Hilbert space H: 1) the algebras of quotients with bounded evaluation of the ideal...

This article deals with bounding sets in real Banach spaces E with respect to the functions in A(E), the algebra of real analytic functions on E, as well as to various subalgebras of A(E). These bounding sets are shown to be relatively weakly compact and the question whether they are always relatively compact in the norm topology is reduced to the study of the action on the set of unit vectors in $l_{\infty }$ of the corresponding functions in $A\left(l_{\infty }\right)$. These results are achieved by studying the homomorphisms on the...

In this paper weare interested in subsets of a real Banach space on which different classes of functions are bounded.

Let A be a locally convex, unital topological algebra whose group of units $A^{\times }$ is open and such that inversion $\iota :A^{\times }\to A^{\times }$ is continuous. Then inversion is analytic, and thus $A^{\times }$ is an analytic Lie group. We show that if A is sequentially complete (or, more generally, Mackey complete), then $A^{\times }$ has a locally diffeomorphic exponential function and multiplication is given locally by the Baker-Campbell-Hausdorff series. In contrast, for suitable non-Mackey complete A, the unit group $A^{\times }$ is an analytic Lie group without...

On étudie certaines algèbres de fonctions analytiques réelles définies sur un ouvert $\Omega$ de ${\mathbf{R}}^n$. La propriété principale de ces algèbres est que tout semi-analytique de $\Omega$ défini globalement à l’aide d’un nombre fini de fonctions de $𝒪\left(\Omega \right)$, admet un nombre fini de composantes connexes. En reprenant les idées de Khovanskii (lemme de Rolle généralisé), on démontre que ces algèbres restent topologiquement noethériennes quand on leur adjoint les solutions de certaines équations différentielles du ler ordre. Par...