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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Nonharmonic Solutions of the Laplace Equation

1. It is known that solutions of the Laplace equation $$\frac{{\partial ^2 {\text{u}}}}{{\partial {\text{x}}^2 }} + \frac{{\partial ^2 {\text{u}}}}{{\partial {\text{y}}^2 }} = 0$$ considered in the space of distributions (hyperfunctions) are always classical solutions called harmonic functions. In this paper we shall consider the Laplace equation in the space of so-called boehmians and show that there may appear solutions which are not classical. The boehmians, we are dealing with, are particular cases of the more general concept of generalized functions introduced in [1], p. 120. Here, they are defined by using delta-sequences.

Jozef Burzyk

### Generalized Functions; Multiplication of Distributions; Applications to Elasticity, Elastoplasticity, Fluid Dynamics and Acoustics

If Ω denotes any open set in ℝn, I have defined an algebra G (Ω) of “generalized functions” on Ω. One has the set of inclusions $${\text{C}}^\infty \left( \Omega \right) \subset D\prime \left( \Omega \right) \subset G\left( \Omega \right)$$ where C∞ (Ω) (respectively D’ (Ω) denotes the set of all C∞ functions (resp. all distributions) on Ω. Two basic points have to be stressed: C∞ (Ω), with its usual pointwise multiplication, is a subalgebra of G(Ω)any element of G(Ω) admits partial derivatives of any order which generalize exactly those in D’ (Ω).

J. F. Colombeau

The paper investigates algebraical and topological structures from a common point of view. The main idea consists in weakening the axioms of an Eilenberg-Moore algebra such that they become natural conditions of a topological structure. By some of these conditions continuous lattices can be characterized.

Werner Gähler

### Simple Applications of Generalized Functions in Theoretical Physics: The Case of Many–Body Perturbation Expansions

Let Ĥ = Ĥ0 + $${\rm{\hat V}}$$ be a self-adjoint operator, bounded from below and defined on a Hilbert space, representing the Himiltonian of an interacting physical system, and Ĥ0 the one for a simpler system with known spectrum and eigenstates. Typically, physicists want to evaluate the (grand–) canonical partition function Tr exp(–βĤ), where β-1 > 0 is Boltzmann’s constant times temperature, and Tr stands for the trace, in powers of $${\rm{\hat V}}$$. For a fixed power of $${\rm{\hat V}}$$, the expansion is unique and consists of a sum of terms, interpreted as physical processes. An individual term can be calculated only if generalized functions are introduced. This is a somewhat arbitrary procedure, however. Different schemes are presented an partial summations of individual terms through all the orders of the expansion in $${\rm{\hat V}}$$ are discussed.

H. F. G. Keiter

### Laplace Transforms of Hyperfunctions: Another Foundation of the Heaviside Operational Calculus

The Laplace transform1$${\rm{\hat f}}\left( \lambda \right) = \int\limits_0^\infty {{\rm{e}}^{ - \lambda {\rm{x}}} {\rm{f}}\left( {\rm{x}} \right){\rm{dx}}}$$is usually defined for a measurable function f(x) on [0,∞ ) satisfying the exponential type condition 2$$\left| {{\rm{f}}\left( {\rm{x}} \right)} \right| \mathop < \limits_ = \,{\rm{C}}\,{\rm{e}}^{{\rm{Hx}}},\quad {\rm{x}} > 0,$$ with constants C and H. Then $${\rm{\hat f}}$$ (λ) is a holomorphic function on the half plane Re λ > H and satisfies the estimates $$\left| {{\rm{f}}\left( \lambda \right)} \right| \mathop < \limits_ = \,{\rm{C}}\,\left( {{\rm{Re}}\lambda - {\rm{H}}} \right)^{ - 1}.$$

Hikosaburo Komatsu

### S-Asymptotic of Distributions

In the last thirty years many definitions of the asymptotoic behaviour of distributions have been presented. We can roughly divide them in two sets. To the first one belong those definitions which directly use the classical definition of the asymptotic behaviour of a numerical function. The distribution T has to be equal to a numerical function f or to a derivative, in the sense of distributions, of a numerical functions, DPf, in a neighbourhood of infinity. The behaviour of the distribution at infinity is in reality the behaviour of the function f or corrected by p. All of these definitions are basically given in the one dimensional case.

Bogoljub Stanković

### The Wiener-Hopf Equation in the Nevanlinna and Smirnov Algebras and Ultra-Distributions

1. The Wiener-Hopf equation on the semi-axis 1.1 $$\phi \left( \xi \right) = \int\limits_0^\infty {{\text{k}}\left( {\xi - \xi \prime} \right)\phi \left( {\xi \prime} \right){\text{d}}\xi \prime + {\text{f}}\left( \xi \right),\quad \xi \mathop > \limits_ = 0}$$ and the associated Riemann-Hilbert problem on a real axis 1.2 $$\rho \left( {\text{x}} \right)\phi ^ + \left( {\text{x}} \right) = \psi ^ - \left( {\text{x}} \right) + {\text{F}}\left( {\text{x}} \right)\quad {\text{a}}{\text{.e}}{\text{.}}\;{\text{on}}\;\mathbb{R}$$ has been investigated by many mathematicians starting from N. Wiener and E. Hopf [1] under various assumptions about kernel k and function ρ. An important contribution to their theory has been made by V. A. Fok [2], N. I. Muschelishvili [3, 4], I. N. Vekua [24], N. P. Vekua [3, 5], V. A. Ambartsumian [6], F. D. Gahov [7], S. Chandrasekhar [8], V. V. Sobolev [9], M. G. Krein [10, 11], I. I. Daniluk [26], B. V. Bojarskii [27], I. B. Simonenko [28], G. S. Litvinchuk [29], M. V. Maslennikov [12], N. B. Engibarjan [13], V. M. Kokilashvili and V. A. Paatashvili [30] and others.

### On Nonlinear Systems of Ordinary Differential Equations

The paper gives some analytical representations and numerical methods for the solutions of systems of ordinary differential equations with emphasis of the formal side, using the connection to the linear partial differential equations in the case first mentioned. The numerical methods are investigated concerning their stability and compared by test calculations.

Lothar Berg

### A New Construction of Continuous Endomorphisms of the Operator Field

In this note we shall give a method to construct continuous endomorphisms of the operator field endowed with convergence structure of type I. The problem is to find, to construct different types of endomorphisms or the linear mapping of the operator field M. In 1967 Gesztelyi published some representation theorems on linear operator transformations, nevertheless there were known only a few types of different transformations. In 1971, in Dubrovnik, R. A. Struble proposed to investigate this problem in view of finding new types of transformations.

A. Bleyer

### Some Comments on the Burzyk-Paley-Wiener Theorem for Regular Operators

The proof of the Paley-Wiener type theorem of Burzyk was outlined by J. Burzyk in his talk during this conference. A full proof is to appear in print soon. The following are some comments concerning his theorem. The theorem can be posed as pertaining to regular operators or as pertaining to regular quotients on the whole line (the Boehmian’s of P. and J. Mikusiński) since those with compact support coincide.

Thomas K. Boehme

### Two Theorems on the Differentiation of Regular Convolution Quotients

We shall discuss two theorems on the derivatives of generalized functions. The class of generalized functions defined below as regular convolution quotients is a generalization of distributions and is also a generalization of the regular Mikusiński operators. Moreover, is a subclass of the quotients defined by J. and P. Mikusiński (Quotients de suites et leurs applications dans l’analyse fonctionnelle, Comptes Rendus, 239, série I (1981)). It is a subclass with some local properties, and we discuss some of these local properties. This subclass has been investigated by Piotr Mikusiński (Convergence of Boehmians, Japan. J. Math., Vol 9 (1983) and Boehmians as generalized functions, to appear Japan. J. Math.).

Thomas K. Boehme

### Values on the Topological Boundary of Tubes

Holomorphic functions in tube domains in ℂn which generalize the Hardy Hp functions are shown to have boundary values on the topological boundary of the tube. The boundary values are obtained restrictedly or unrestrictedly in Lp and in the distribution space $${\text {K}}_{1}^{\prime}$$.

Richard D. Carmichael

### Abelian Theorem for the Distributional Stieltjes Transformation

We study the behaviour of the distributional Stieltjes transformation (Srf) (z), z ∈ ℂ \ ℝ, at zero of an f ∈ S’ which has the appropriate quasi-asymptotic behaviour at zero. By using the known results for the asymptotic behaviour at ± ∞, we obtain a final value Abelian theorem for the distributional Stieltjes transformation at zero.

Danica Nikolić-Despotović, Stevan Pilipović

### Some Results on the Neutrix Convolution Product of Distributions

The convolution product of two distributions is normally defined as folows, see Gelfand and Shilov [3].

Brian Fisher

### On Generalized Transcedental Functions and Distributional Transforms

During the last two and half decades, Meijer C.S. [l] G-function and its generalization in one variable due to Fox C. [2] by the symbol H in two variables by Agarwal R.P. [3], Sharma B.L. [4], Mourya D.P. [5] in n-variables by Khadiya S.S. and Goyal A.N. [6] and their respective representations in H-symbol of two and n-variables have given great impetus to researches in special functions. However, J. Gopal Krishana and Muhammed Ghouse [7]; Buschman R.G. [8]; Tandon O.P. [9] have raised certain questions regarding the path of integration and existence in the case of G and H functions of two variables.

A. N. Goyal, V. K. Chaturvedi

### An Algebraic Approach to Distribution Theories

In this note Mikusiński’s idea of convolution quotients is generalized in two directions simultaneously: – The ring R is not merely acting on itself but acts also on a separate vector space V.– The ring R need not be commutative.This approach makes it possible to bring many theories of generalized functions under the same viewpoint. So e.g. Schwartz’ tempered distribution space, many (all?) distribution spaces of Gelfand-Šilov and Mikusiński’s space of convolution quitients (cf. the Examples) are all very special examples of the construction that we present here. The authors expect that many more locally convex topological vector spaces can be brought under the same viewpoint. The connection with (non-commutative) harmonic analysis, especially the case that R is a (subset of) the convolution algebra of a Lie group is now being studied by the second author. The algebra of this paper is inspired by Ore’s construction of non-commutative fields. See [5], p. 119.

J. de Graaf, A. F. M. ter Elst

### Products of Wiener Functionals on an Abstract Wiener Space

Mikusiński in [1] has proved that the product of the distributions δ (x) and pf. $$\frac{1} {{\text{x}}}$$ on the one-dimensional Euclidean space ℝ exists in the sense of generalized operations and equals $$- \frac{1} {{\text{2}}}\delta \prime \left( {\text{x}} \right)$$ . This result can be easily extended to the case of an n-dimensional Euclidean space ℝn, i.e. for any $$\ell = \left( {\ell _1,\ell _2, \ldots,\ell _{\text{n}} } \right) \in R^{\text{n}},\left( {\ell \ne 0} \right)$$ , $$\delta \left( {\left( {\ell,{\text{x}}} \right)} \right) \cdot {\text{pf}}{\text{.}}\frac{1} {{\left( {\ell,{\text{x}}} \right)}} = - \frac{1} {2}\delta \prime \left( {\left( {\ell,{\text{x}}} \right)} \right)\quad {\text{x}} = \left( {{\text{x}}_{\text{1}}, \ldots,{\text{x}}_{\text{n}} } \right) \in R^{\text{n}},$$ where $$\left( {\ell,{\text{x}}} \right) = \sum\limits_{{\text{k}} = 1}^{\text{n}} {\ell _{\text{k}} {\text{x}}_{\text{k}}}$$ .In this paper we shall try to extend the above results to the case of an infinite dimensional space i.e. an abstract Wiener space.

Shiro Ishikawa

### Convolution in K’ {Mp}-Spaces

I. M. Gelfand and G. E. Shilov introduced in [3] (see p. 78) spaces of generalized functions, dual to the spaces K{Mp}defined by means of an arbitrary non-decreasing sequence {Mp} of functions Mp : ℝd → [1,∞], which are supposed to be continuous on the set S = Sp = {x ∈ ℝd : Mp(x) > ∞} (p ∈ N).

A. Kamiński, J. Uryga

### The Problem of the Jump and the Sokhotski Formulas in the Space of Generalized Functions on a Segment of the Real Axis

Let Sm,n (m,n are fixed, m ≧ 0, n ≧ 0) denote the linear countable normed space of smooth functions that can be represented with their derivatives in the form 1 $$\phi ^{\left( {\text{k}} \right)} \left( {\text{t}} \right) = \frac{{\phi _{\text{k}}^0 \left( {\text{t}} \right)\ln ^{\ell _{\text{k}} } \left( {{\text{t}} - {\text{a}}} \right)\ln ^{{\text{q}}_{\text{k}} } \left( {{\text{b}} - {\text{t}}} \right)}} {{\left( {{\text{t}} - {\text{a}}} \right)^{{\text{m}} + \alpha _{\text{k}} + {\text{k}}} \left( {{\text{b}} - {\text{t}}} \right)^{{\text{r}} + {{\beta }}_{\text{k}} + {\text{k}}} }},{\text{k}} = 0,1,2, \ldots ,$$ where 0 ≦ αk < 1, 0 ≦ βk < l, ℓk,qk ≧ 0, $$\phi _{\text{k}}^0 \left( {\text{t}} \right)$$ (k = 0,1,2,…)are smooth functions on (a,b) and H-continuous on [a,b]; a function ψ is an H-function or Hölder’s function, from Hλ, λ > 0, if there is a constant A so that $$\left| {\psi \left( {{\text{t}}_{\text{1}} } \right) - {{\psi }}\left( {{\text{t}}_{\text{2}} } \right)} \right| < {\text{A}}\left| {{\text{t}}_{\text{1}} - {\text{t}}_2 } \right|^\lambda$$ for all t1, t2 ∈ [a, b].

L. V. Kartashova, V. S. Rogozhin

### A Generalized Fractional Calculus and Integral Transforms

In this paper a generalized fractional calculus and its applications to different topics in analysis, especially to some integral transforms, are discussed. The kernel-function of the generalized operators of integration of fractional multiorder considered here is a suitably chosen case of Meijer’sG-function: 1 $${\text{G}}_{{\text{pq}}}^{{\text{mn}}} \left[ {\sigma \left| {\begin{array}{*{20}c} {{\text{a}}_{\text{1}} , \ldots ,{\text{a}}_{\text{p}} } \\ {{\text{b}}_{\text{1}} , \ldots ,{\text{b}}_{\text{q}} } \\ \end{array} } \right.} \right] = \frac{1} {{2\pi {\text{i}}}}\int\limits_L {\frac{{\prod\limits_{{\text{k}} = {\text{1}}}^{\text{m}} {\Gamma \left( {{\text{b}}_{\text{k}} - {\text{s}}} \right)} \prod\limits_{{\text{j}} = {\text{1}}}^{\text{n}} {\Gamma \left( {{\text{1}} - {\text{a}}_{\text{j}} + {\text{s}}} \right)} }} {{\prod\limits_{{\text{k}} = {\text{m}} + {\text{1}}}^{\text{q}} {\Gamma \left( {1 - {\text{b}}_{\text{k}} + {\text{s}}} \right)} \prod\limits_{{\text{j}} = {\text{n}} + {\text{1}}}^{\text{p}} {\Gamma \left( {{\text{a}}_{\text{j}} - {\text{s}}} \right)} }}} \sigma ^{\text{s}} {\text{ds}}\quad \left( {\left[ {\text{1}} \right],\left[ 2 \right]} \right).$$

Virginia Kiryakova

### On the Generalized Meijer Transformation

Following the method of Mikusiński [l], Ditkin [2] and later with Prudnikov [3] developed an operational calculus for the operator DtD. In the 60’s, Meller [4] generalized Ditkin’s calculus to the operator Bα = t-α Dt1+αD with α ∈ (-1,1). Generalizations to Bessel operators of a higher order were made by Botashev [5], Dimovski [6], Krätzel [7] and others. Koh [8] extended Meller’s results to α > 1 by using fractional calculus. Later [9], a direct extension was achieved in which the convolution of Ф(t) and ψ(t) is given by 1$$\phi *\psi = \frac{1}{{\Gamma \left( {\alpha + 1} \right)}}{D^\alpha }DtD\int\limits_0^t {\int\limits_0^1 {{\eta ^\alpha }{{\left( {1 - x} \right)}^\alpha }\phi \left( {x\eta } \right)\left[ {\psi \left( {1 - x} \right)\left( {t - \eta } \right)} \right]dxd\eta ,} }$$ where α ≥ 0, Dα = DnIn-α and Iv is the Riemann-Liouville integral, 2 $${\text{I}}^v {\text{f}}\left( {\text{t}} \right) = \frac{1} {{\Gamma \left( v \right)}}\int\limits_0^{\text{t}} {\left( {{\text{t}} - \xi } \right)^{v - 1} {\text{f}}\left( \xi \right){\text{d}}\xi {\text{.}}}$$

E. L. Koh, E. Y. Deeba, M. A. Ali

### The Construction of Regular Spaces and Hyperspaces with Respect to a Particular Operator

Let H be a Hilbert space with an inner product (·,·) and corresponding norm ║·║. There is given an unboundedoperatorB : D(B) ⊂ H → H such that 0 ∈ (B) and – B is a generator of an analytic semigroup. That is, $$\text{p}(\text{B}) \supset \sum {^ + = \{ \lambda |\lambda \in {\not {\text C}},\,0 < \omega < |\arg \lambda |\mathop < \limits_ = \pi \} \cup \{ 0\} }$$ and $$\left\| {{\text{R}}\left( {\lambda ;{\text{B}}} \right)} \right\| {\mathop < \limits_ = } \frac{{\text{M}}} {\lambda },\forall \lambda \in \sum ^ +$$ where ω < π/2 and M is a positive constant, see, e.g., [4]. Note that, in this case, B* also satisfies conditions (1.1) and (1.2), hence – B* also generates an analytic semigroup e–tB* = (e–tB)*. For such an operator B, powers of arbitrary order can be defined an they enjoy nice properties. We shall collect some of them as follows (Conf., e.g., [4])

G. Liu

### Operational Calculus with Derivative Ŝ = S2

Boundary value problems for abstract differential equations were considered among others in [7], [12]. In [10] the author has constructed an operational calculus by operation sq S and by linear operation B : L2 → Ker S under some special assumptions (see [10], p. 252). This paper is a generalization of some conclusions presented in [10].

Eligiusz Mieloszyk

### Solvability of Nonlinear Operator Equations with Applications to Hyperbolic Equations

Consider nonlinear equations of the form 1 $${\text{Au}} - {\text{F}}\left( {{\text{x,}}\,{\text{u}}} \right) = {\text{f}}\left( {\text{x}} \right),\,{\text{x}} \in {\text{Q}}$$ in H = L2(Q,ℝm), where Q is a bounded domain in ℝn, f ∈ H is given, F: Q × ℝm → ℝm is a Caratheodory function and A : D(A) ⊂ H → H is a selfadjoint map with possibly ∞-dimensional null space.

P. S. Milojević

### Some Important Results of Distribution Theory

The theme of the conference entitled ‘Generalized Functions Convergence, Structures and their Applications’ is based on the vital mathematical theory of Functional Analysis envolved in present century, hailed by Browder (1972) as the century of functional analysis. The opinion of some mathematicians about functional analysis as a purely mathematical abstraction is true to some extent. In response to this question, Dieudonne (1972) has referred to the applications of the theory of distributions created by the French mathematician Laurent Schwartz since his theory is based on functional analysis, but further work is necessary, keeping in mind the historical perspective pointed out by Dieudonne.

O. P. Misra

### Hyperbolic Systems with Discontinuous Coefficients: Examples

Consider the initial value problem for a linear hyperbolic (n×n)-system in two variables1 $$\begin{array}{*{20}l}(\partial_{t}\ +\ \Lambda(x, t)\partial_{X})V\ =\ F(x,t)V\ +\ G(x,t),(x,t) \in IR^2\\V(x,0)\ =\ A(X), x\ \in\ IR\end{array}$$ where Λ and F are (n×n)-matrices, Λ real valued and diagonal, and V, G, A are n-vectors.

Michael Oberguggenberger

### Estimations for the Solutions of Operator Linear Differential Equations

In this paper we observe the approximate solution of the linear operator differential equation and estimate the error of approximation. For this purpose we use the results from [6]. They enable us to introduce some measures of approximation on the space L of locally integrable functions on [0,∞) and on the field of Mikusiński operators.

Endre Pap, Đurđica Takači

### Invariance of the Cauchy Problem for Distribution Differential Equations

Let n > 2 be an integer. In Persson [2] and [3], the Cauchy problem for the equation 1.1 $${\text{u}}^{\left( {\text{n}} \right)} + {\text{a}}_{{\text{n}} - 1} {\text{u}}^{\left( {{\text{n}} - 1} \right)} + \ldots + {\text{a}}_{\text{0}} {\text{u}} = {\text{f}}$$ is treated. The new thing is that some coefficients and f are allowed to be distributions and not necessarily measures. Then some of the derivatives u(j), 0 ≦ j < n, may not be pointwise defined. Still, a Cauchy problem can be defined for (1.1) with n initial data as in the ordinary Cauchy problem for measure differential equations. The function u is defined as a solution of an integral equation. Here, iterated primitive distributions of some of the coefficients are involved. One also chooses an iterated primitive distribution of f. As long as the primitive distribution is not pointwise defined, one makes a choice differing from another choice by a constant. In case of iterated primitive distributions, the difference is a polynomial. As soon as the primitive distribution is pointwise defined, one chooses the primitive distribution to be zero at the initial point of the Cauchy problem, just as one does in the measure differential equation case. We prove that the affine space of solutions of (1.1) is invariant under the choice of the iterated primitive distributions of the coefficients and of f.

### On the Space , q ∈ [1,∞]

We shall present in this paper mainly the results for q = 2. Namely, we shall present the results from [3 – 7] concerning the space of Beurling ultradistributions $$\upsilon _{{\text{L}}^2 }^{'\,^{\left( {{\text{M}}_{\text{p}} } \right)} }$$ . In our investigations we follow the Komatsu approach to spaces of ultradistrinutions [2], so for the notions and the basic results of ultradistribution theory we refer the reader to this paper.

S. Pilipović

### Peetre’s Theorem and Generalized Functions

Sheaf morphisms are considered in sheaves of generalized functions. It is proved that for (ultra)distributions they must be continuous outside discrete points. Contrary to Peetre’s original theorem, which applies to sheaves of test functions, an example makes clear that these points can really be points of discontinuity. Finally, it is shown that in the sheaf of hyper-functions there are more general discontinuous sheaf morphisms.Peetre’s theorem says that any sheaf morphism in the sheaf of C∞-functions is a differential operator. We shall investigate sheaf morphisms in sheaves of generalized functions, in particular distributions, ultradistributions of the Beurling and of the Roumieu type, and hyperfunctions. All these sheaves are soft so that their sections with a compact support form flabby cosheaves which are the duals, with respect to a certain topology, of the sheaves of their associated test functions. The main point is to investigate the continuity of a cosheaf morphism P (= local operator) in one of these cosheaves. At places where P is continuous its transposed tP is a continuous sheaf morphism in the sheaf of test functions and it follows that tP, and hence P itself, are appropriate differential operators there. In this paper we shall only briefly mention these results, as well as the generalization of Peetre’s theorem to the soft sheaves of test functions. Our main attention will be on the continuity of a local operator in a space of generalized functions and we shall indicate what possibilities there are for a discontinuous sheaf morphism.

J. W. de Roever

### Infinite Dimensional Fock Spaces and an Associated Generalized Laplacian Operator

Quantum field theory has a long history of cooperation between physicists and mathematicians. The approach in this paper utilizes the theory developed in generalized functions. This technique was pioneered by Dr. Paul Dirac. His results of the 1920’s still remain a very elegant treatment of the theory. A recent publication [9] written by Dirac in 1966 gives a concise assessment of the subject. Many other contributors such as Bergmann, Bogoliubov, Cholewinski, Colombeau, Friedrichs, Kastler, Kristensen, Mejblo, Poulsen, Rzewuski, Schiff, Shapiro, and Wightman incorporate generalized functions into their development.

John Schmeelk

### The n-Dimensional Stieltjes Transformation

In a previous paper ([8]), we analysed the relation between Silva’s order of growth (introduced in [3]) with equivalence at infinity of distributions ([3]) and applied them to the distributional Stieltjes transformation (see [5]).In this paper we shall define the n-dimensional versions of these notions, and, in particular, we shall prove some Abelian theorems for the n-dimensional Stieltjes transformation.

### Colombeau’s Generalized Functions and Non-Standard Analysis

Using some methods of Non-Standard Analysis we modify one of Colombeau’s classes of generalized functions. As a result we define a class Ê of so-called metafunctions which possesses all the good properties of Colombeau’s generalized functions, i.e. (i) Ê is an associative and commutative algebra over the system of so-called comptex meta-numbers ℂ̂(ii) Every meta-function has partial derivatives of any odrer (which are meta-functions again); (iii) Every meta-function is integrable on any compact set of ℝn and the integral is a number from ℂ̂ (iv) Ê contains all the tempered distributions S’, i.e. S’ ⊂ Ê isomorphically with respect to all the linear operations (including the differentiation). Thus, within the class Ê the problem of multiplication of the tempered distributions is satisfactorily solved (every two distributions in S’ have a well-defined product in Ê). The crucial point is that ℂ̂ is a field in contrast to the system of Colombeau’s generalized numbers ℂ̂ which is a ring only ℂ̂ is the counterpart of ℂ̂ in Colombeau’s theory). In this way we simplify and improve slightly the properties of the integral and the notion of “values of the meta-functions”, as well as the properties of the whole class Ê itself if compared with the original Colombeau theory. And, what is maybe more important, we clarify the connection between Non-Standard Analysis and Colombeau's theory of new generalizedfunction in the framework of which the problem of the multiplication of distributions was recently solved.

T. D. Todorov

### One Product of Distributions

In this paper we prove the inequality by which the Fourier transform of the product |x|r s(x) for s ∈ S(ℝn), x ∈ ℝn, r > 0, is estimated. The product $${\left| {\text{x}} \right|^{\text{r}}}\cdot\mathop f\limits^ \wedge$$ is also defined, where f is a real locally integrable function on ℝn and $$\mathop f\limits^ \wedge$$ it is the Fourier transforms and its proved that this product is a tempered distribution.

Miloš Tomić

### Abel Summability for a Distribution Sampling Theorem

Let F(w)be an L2 function with compact support on [-σ,σ]; let T = π/σ and f(t) be the Fourier transform of F(w). Then the well-known sampling theorem says $${\text{f}}\left( {\text{t}} \right) = \sum\limits_{{\text{n}} = - \infty }^\infty {{\text{f}}\left( {{\text{nT}}} \right)\frac{{\sin \sigma \left( {{\text{t}} - {\text{nT}}} \right)}} {{\sigma \left( {{\text{t}} - {\text{nT}}} \right)}}},$$ where convergence is uniform in ℝ1. If F(w) is now a distribution with compact support on [-σ,σ] the Fourier transform is still a function but the series does not converge necessarily. However it is shown, under mild conditions of F(w), that the series is Abel summable, i.e. $${\text{f}}\left( {\text{t}} \right) = \mathop {\lim }\limits_{{\text{r}} \to l^ - } \sum\limits_{{\text{n}} = - \infty }^\infty {{\text{r}}^{\left| {\text{n}} \right|} \,{\text{f}}\left( {{\text{nT}}} \right)\frac{{\sin \sigma \left( {{\text{t}} - {\text{nT}}} \right)}} {{\sigma \left( {{\text{t}} - {\text{nT}}} \right)}}}$$ where the convergence is uniform on bounded sets in ℝ1.

Gilbert G. Walter

### On the Value of a Distribution at a Point

We show connections between the notion of the value of a distribution at a point in the Łojasiewicz sense and the integrability of its Fourier transform. We consider the one-dimensional case.

Ryszard Wawak

### On Interchange of Limits

Using matrix methods we prove theorems on interchange of limits for matrices (double sequences) whose elements are in an abelian group equiped with a convergence. Proofs of theorems on interchange of limits, uniform convergence, equicontinuity, uniform countable additivity, uniform boundedness can be reduced to the problem of convergence to zero of diagonals of certain matrices, so-called K-matrices (see, [l], [2]).

Piotr Antosik

### Countability, Completeness and the Closed Graph Theorem

The webs of M. De Wilde [4] have made an enormous contribution to the closed graph theorems in locally convex spaces(lcs). Although webs have a very intricate layered construction, two properties in particular have contributed to the closed graph theorem. First of all, webs possess a strong countability condition in the range space which suitably matches the Baire property of Fréchet spaces in the domain space; as a result the zero neighbourhood filter is mapped to a p-Cauchy filter, a filter attempting to settle down. Secondly webs provide a completeness condition which allow p-Cauchy filters to converge.

R. Beattie, H.-P. Butzmann

### Inductive Limits of Riesz Spaces

The inductive limit of a family of Riesz spaces is introduced and investigated.

Wolfgang Filter

### Convergence Completion of Partially Ordered Groups

As is well known, the (MacNeille, [7]) conditional completion by nonvoid cuts of a partially ordered group (which is the unique conditionally complete lattice each element of which is a join and a meet of the group’s elements) cannot in general (in fact whenever the group fails to be “Archimedean”) be made into a partially ordered group. There is a largest subset of the completion to which the group composition can be extended so as to achieve a partially ordered group (see Fuchs [6] or below). In the totally ordered case this subset may be attained intrinsically as the completion of the original group in its order-topology (Cohen-Goffman [4]). There is no suitable order-topology even for lattice-ordered groups; one can use certain down-directed subsets of positive elements with meet zero to induce order-theoretic topological group structures whose completions may then be shown to be canonically contained in the MacNeille completion (Banaschewski [1]); and more general such down-directed subsets to induce order-theoretic non-group convergence structures whose completions are also so contained - indeed, the totality of these suffice to attain the largest group subextension of the MacNeille completion of a commutative lattice-ordered group (Ibid.). This procedure has been identified as completion with respect to a form of order-convergence by Papangelou [8], who is then able to give a much more efficient proof of the same result. To extend this to partially ordered (possibly non-commutative) groups, it is necessary to isolate the appropriate notion of order-convergence and to devise a proof independent of the more special properties this notion has in l-groups. The result is to make every partially ordered group into a convergence group whose convergence completion is exactly the largest possible partially ordered group in the conditional (order) completion.

Isidore Fleischer

### Some Results from Nonlinear Analysis in Limit Vector Spaces

In paper [10] Bieri Hanspeter obtained some results from nonlinear analysis in limit vector spaces. Using a variant of the KKM lemma in limit vector spaces, we shall prove in this paper some fixed point theorems in limit vector spaces. A generalization of the Ky Fan minimax principle in limit vector spaces is also obtained.

### Completions of Cauchy Vector Spaces

T2 and T3 completions of Cauchy vector spaces are studied. Every T2 Cauchy vector space is shown to have a strict T2 Cauchy vector space completion. A Cauchy vector space has a T3 Cauchy vector space completion exactly when the underlying Cauchy space has a T3 Cauchy space completion.

D. C. Kent, G. D. Richardson

### Regular Inductive Limits

Given a sequence E1 ⊂ E2 ⊂ … of locally convex spaces with continuous inclusions, the locally convex inductive limit E = indlim En is called regular if every set bounded in E is also bounded in some En. Several necessary and sufficient conditions for the regularity of E are derived.

Jan Kucera

### Weak Convergence in a K-Space

The notions of K-convergence and K-space have proved to be very useful in functional analysis. In monographs [2] and [3] there are many results on these subjects and we shall use the notations and notions from them.

Endre Pap

### The Banach-Steinhaus Theorem for Ordered Spaces

Let X and Y be vector lattices and Ti: X → Y a sequence of linear operators which are sequentially continuous with respect to relative uniform convergence. If {Tjx} is relatively uniformly convergent to Tx for each x ∈ X, under appropriate assumptions on the spaces, we show that the linear operator T is also continuous and that the {Ti} are order equicontinuous in a certain sense. We also establish an order version of the Uniform Boundedness Principle.

Charles Swartz

### Open Problems

Assume that f ∈ D’, g ∈ C, g(x0 ≠ 0 and $$\left( {f\; * \;\delta _n } \right)\;\left( {g,\tilde \delta _n } \right)\;\mathop \to \limits^{D'} \;0$$ for every delta-sequences $$\delta _n \;and\;\tilde \delta _n \;.$$

P. Antosik, J. Burzyk

### Backmatter

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