Generalized Functions, Convergence Structures, and Their Applications

This Proceedings consists of a collection of papers presented at the International Conference "Generalized functions, convergence structures and their applications" held from June 23-27, 1987 in Dubrovnik, Yugoslavia (GFCA-87): 71 participants from 21 countr~es from allover the world took part in the Conference. Proceedings reflects the work of the Conference. Plenary lectures of J. Burzyk, J. F. Colombeau, W. Gahler, H. Keiter, H. Komatsu, B. Stankovic, H. G. Tillman, V. S. Vladimirov provide an up-to-date account of the cur­ rent state of the subject. All these lectures, except H. G. Tillman's, are published in this volume. The published communications give the contemporary problems and achievements in the theory of generalized functions, in the theory of convergence structures and in their applications, specially in the theory of partial differential equations and in the mathematical physics. New approaches to the theory of generalized functions are presented, moti­ vated by concrete problems of applications. The presence of articles of experts in mathematical physics contributed to this aim. At the end of the volume one can find presented open problems which also point to further course of development in the theory of generalized functions and convergence structures. We are very grateful to Mr. Milan Manojlovic who typed these Proce­ edings with extreme skill and diligence and with inexhaustible patience.

Table of Contents:
Section I. Plenary Lectures.- Nonharmonic solutions of the Laplace equation.- Generalized functions; multiplication of distributions; applications to elasticity, elastoplasticity, fluid dynamics and acoustics.- Monads and convergence.- Simple applications of generalized functions in theoretical physics: the case of many-body perturbation expansions.- Laplace transforms of hyperfunctions: another foundation of the Heaviside operational calculus.- S-asymptotic of distributions.- The Wiener-Hopf equation in the Nevanlinna and Smirnov algebras and ultradistributions.- Section II. Generalized Functions.- On nonlinear systems of ordinary differential equations.- A new construction of continuous endomorphisms of the operator field.- Some comments on the Burzyk-Paley-Wiener theorem for regular operators.- Two theorems on the differentiation of regular convolution quotients.- Values on the topological boundary of tubes.- Abelian theorem for the distributional Stieltjes transformation.- Some results on the neutrix convolution product of distributions.- On generalized transcedental functions and distributional transforms.- An algebraic approach to distribution theories.- Products of Wiener functionals on an abstract Wiener space.- Convolution in K’{Mp}-spaces.- The problem of the jump and the Sokhotski formulas in the space of generalized functions on a segment of the real axis.- A generalized fractional calculus and integral transforms.- On the generalized Meijer transformation.- The construction of regular spaces and hyperspaces with respect to a particular operator.- Operational calculus with derivative ? = S2.- Solvability of nonlinear operator equations with applications to hyperbolic equations.- Some important results of distribution theory.- Hyperbolic systems withdiscontinuous coefficients: examples.- Estimations for the solutions of operator linear differential equations.- Invariance of the Cauchy problem for distribution differential equations.- On the space $$\upsilon _{{\text{L}}^{\text{q}} }^{\prime\,^{\left( {{\text{M}}_{\text{p}} } \right)} } $$ , q ? [1, ?].- Peetre’s theorem and generalized functions.- Infinite dimensional Fock spaces and an associated generalized Laplacian operator.- The n–dimensional Stieltjes transformation.- Colombeau’s generalized functions and non-standard analysis.- One product of distributions.- Abel summability for a distribution sampling theorem.- On the value of a distribution at a point.- Section III. Convergence Structures.- On interchange of limits.- Countability, completeness and the closed graph theorem.- Inductive limits of Riesz spaces.- Convergence completion of partially ordered groups.- Some results from nonlinear analysis in limit vector spaces.- Completions of Cauchy vector spaces.- Regular inductive limits.- Weak convergence in a K-space.- The Banach-Steinhaus theorem for ordered spaces.- Section IV. Open Problems.- Open problems.- Participants.