J.F. Colombeau

A New Theory of Generalized Functions

Publisher Summary This chapter presents a new theory of generalized functions. It reveals that the distribution theory has two kinds of defects. It does not give a general multiplication of distributions. This is very important since the computations of Quantum Field Theory are based upon multiplications of distributions. In the same viewpoint let us quote that the restriction of a distribution to a linear subspace is not defined in general, as well as the composition of distributions. It considers a linear partial differential equation with non-constant coefficients. To circumvent these defects various kinds of generalized functions, more general than distributions, were introduced but they always have the above defects.

Holomorphic and Differentiable Mappings of Uniform Bounded Type

We prove that each holomorphlc or Cinfin; mapping from a (DFM)-space into a Frechet space is of uniform hounded type. This result unifies and clarifies the relationships between several known results, hitherto apparently unrelated, in topics as varied as convolution equations, Hahn-Banach extensions of holomorphic mappings, and the δ-equation in locally convex spaces.

Convolution equations in spaces of infinite dimensional entire functions

Abstract We prove general results of surjectivity for convolution equations in spaces of entire functions in locally convex spaces. These results improve and partially unify known results due to Berner, Boland, Dwyer, Gupta and Matos. We also obtain results on approximation of solutions.

A Result of Existence of Holomorphic Maps Which Admit a Given Asymptotic Expansion

Publisher Summary This chapter presents a study on a result of existence of holomorphic maps, which admits a given asymptotic expansion. The formal perturbation series of Quantum Fields are believed to be asymptotic expansims of well-defined maps and this motivates to study the asymptotic expansions in infinitely many dimensions. An infinite dimensional version of a classical theorem of E. Bore1 (which asserts the existence of C ∝ -functions having a given infinite sequence of successive derivatives at a given point) is proved. The chapter presents an investigation on the complex case and the quite remarkable result is obtained that in some usual infinite dimensional spaces, given any formal power. A Silva space is a strong dual of a Frechet–Schwartz space.

Infinite Dimensional Holomorphic “Normal Forms” of Operators on the Fock Spaces of Boson Fields and an Extension of the Concept of Wick Product

Abstract It is a widespread idea among Theoritical Physicists to represent the Fock spaces of Boson fields by suitable spaces of infinite dimensional holomorphic functions (this idea is, for example, contained in a classical book by F.A. Berezin [1]) and to write down, in the form of infinite dimensional holomorphic functions, the so-called “operators representable in the normal form”. We present here a mathematical study of these ideas: we show that “most” of the operators on the Fock spaces of Boson fields are characterized by their “normal form” defined in section 4 by means of a formula which plays a fundamental role in the applications. In a more abstract point of view and via the (infinite dimensional) Fourier-Borel transform, the concept of normal form is very closely related to some sorts of “Kernel-theorems” in infinite dimensional Holomorphy ([8]). The results presented here were announced in two Notes [5][6]; a particular case was obtained independently in [14].

Finite-Difference Partial Differential Equations in Normed and Locally Convex Spaces

Abstract We prove existence of C ∞ -solutions u of equations Du = f, when D is a finite-difference linear partial differential operator with constant coefficients and f is a C ∞ -function defined on a locally convex space, which extends a classical result of Ehrenpreis in the finite dimensional case. The main difficulty in this extension came from the Paley-Wiener-Schwartz theorem in infinite dimension. We also obtain new results for some convolution equations in H (E) when E is a complex space.

Strong Nuclearity in Spaces of Holomorphic Mappings

Publisher Summary This chapter presents a study on strong nuclearity in spaces of holomorphic mappings. Strong nuclearity of spaces of holornorphic are characterized, respectively Silva holomorphic, mappings on open subsets in locally convex and convex bornological, spaces. The space of all holomorphic functions on an arbitrary open subset of a strong dual of a strongly nuclear (F)-space is a strongly nuclear (F)-space under the compact open topology. All the proofs rely on the fact that a linear mapping between two normed spaces is strongly nuclear if and only if for every natural number “n” it can be represented as a composition of n nuclear mappings. The first proof consisted in a reduction to Bolands nuclearity result. The proof, which is presented, is a nice application of the nuclear bornology of a Banach space. Throughout the chapter, all locally convex (l.c) spaces are assumed to be Hausdorff and to be complex vector spaces, and a bornological vector (b.v.) space always denotes a complex, convex, separated, and complete bornological vector space in the terminology of Hogbe–Nlend.

Convolution Equations in Infinite Dimenstions: Brief Survey, New Results and Proofs

Abstract In the last fifteen years a large amount of results were obtained on convolution equations in normed and locally convex spaces. The aim of this work is to contribute to the improvement and clarification of the theory by presenting new results and connections between previously known theorems. For convenience and necessity of presentation we recall most existing results and give their references, so that this paper is also a brief survey on the subject.