Pontryagin spaces of entire functions I

We give a generalization of L.de Branges theory of Hilbert spaces of entire functions to the Pontryagin space setting. The aim of this-first-part is to provide some basic results and to investigate subspaces of Pontryagin spaces of entire functions. Our method makes strong use of L.de Branges's results and of the extension theory of symmetric operators as developed by M.G.Krein.     

Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces

In this paper we give global characterisations of Gevrey–Roumieu and Gevrey–Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace–Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their α-duals in the sense of Köthe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.     

Generalized resolvent matrices and spaces of analytic functions

We introduce triplet spaces for symmetric relations with defect index (1, 1) in a Pontryagin space. Representations of Pontryagin spaces by spaces of vector-valued analytic functions are investigated. These concepts are used to study 2×2-matrix valued analytic functions which satisfy a certain kernel condition.     

Division problem and partial differential equations with constant coefficients in Colombeau's space of new generalized functions

We study the problem of division in Colombeau's space of new generalized functions in the associated sense: more precisely we solve the equation of the formF·G _p H with adequate assumptions onF. By using the Fourier transformation we construct, by simple methods, the solution in the p-associated sense of a partial differential equation with constant coefficients.     

Some functional equations in the space of uniform distribution functions

In this paper various functional equations which arise in the study of binary operations on the set of uniform probability distribution functions are considered and solved.     

Distributional method for the D′ Alembert equation

We consider the D′ Alembert equation in the space of Schwartz distributions and as an application we find the locally integrable solutions of the equation.Received: 22 July 2004; revised: 12 October 2004     

Some remarkable lines of triangles in real normed spaces and characterizations of inner product structures

Summary In this work we consider the heights and the bisectrices of a triangle in a real normed space. Using well-known formulas which can be generalized to real normed spaces we obtain a collection of new characterizations of inner product spaces.     

Strong solutions for differential equations in abstract spaces

Let (E,F) be a locally convex space. We denote the bounded elements of E by . In this paper, we prove that if B_Eb is relatively compact with respect to the F topology and f:I×E_b→E_b is a measurable family of F-continuous maps then for each x₀∈E_b there exists a norm-differentiable, (i.e. differentiable with respect to the ∥·∥_F norm) local solution to the initial valued problem u_t(t)=f(t,u(t)), u(t₀)=x₀. All of this machinery is developed to study the Lipschitz stability of a nonlinear differential equation involving the Hardy-Littlewood maximal operator.     

Regularization and general methods in the theory of functional equations

Summary The aim of this paper is to give a survey of two fields of the theory of non-iterated functional equations with two variables. One is the application of new, general methods of functional analysis, harmonic analysis and other topics to get a unified treatment of several kinds of equations. The other includes general regularity results for non-iterated functional equations with two variables.     

Stability of exponential equations in Schwartz distributions

We reformulate the superstability of exponential equation and cosine functional equation [J.A. Baker, The stability of cosine equation, Proc. Amer. Math. Soc. 80 (1980) 411–416] in some spaces of generalized functions such as the Schwartz distributions, Sato hyperfunctions, and Gelfand generalized functions, which completes the previous results of partial generalizations of the stability problems [J. Chung, A distributional version of functional equations and their stabilities, Nonlinear Anal. 62 (2005) 1037–1051; J. Chung, S.Y. Chung, D. Kim, The stability of Cauchy equations in the space of Schwartz distributions, J. Math. Anal. Appl. 295 (2004) 107–114].     

Multiplicative symmetry and related functional equations

Summary Let (G, *) be a commutative monoid. Following J. G. Dhombres, we shall say that a functionf: G G is multiplicative symmetric on (G, *) if it satisfies the functional equationf(x * f(y)) = f(y * f(x)) for allx, y inG. (1)Equivalently, iff: G G satisfies a functional equation of the following type:f(x * f(y)) = F(x, y) (x, y G), whereF: G × G G is a symmetric function (possibly depending onf), thenf is multiplicative symmetric on (G, *).In Section I, we recall the results obtained for various monoidsG by J. G. Dhombres and others concerning the functional equation (1) and some functional equations of the formf(x * f(y)) = F(x, y) (x, y G), (E) whereF: G × G G may depend onf. We complete these results, in particular in the case whereG is the field of complex numbers, and we generalize also some results by considering more general functionsF. In Section II, we consider some functional equations of the formf(x * f(y)) + f(y * f(x)) = 2F(x, y) (x, y K), where (K, +, ·) is a commutative field of characteristic zero, * is either + or · andF: K × K K is some symmetric function which has already been considered in Section I for the functional equation (E). We investigate here the following problem: which conditions guarantee that all solutionsf: K K of such equations are multiplicative symmetric either on (K, +) or on (K, ·)? Under such conditions, these equations are equivalent to some functional equations of the form (E) for which the solutions have been given in Section I. This is a partial answer to a question asked by J. G. Dhombres in 1973.