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The aim of this survey talk is to show that recently functional equations in a single variable, also called iterative functional equations, turned out to be useful as an elementary and handful tool to study peculiar functions, in particular, continuous nowhere differentiable (cnd) functions.
Abstract: We improve and extend the Ekeland variational principle weakening simultaneously its assumptions. In particular, we prove a geometrical result which combines the Ekeland result with the classical theorem of Weierstrass.
Abstract. This paper provides isomorphisms of a space of analytic functions onto spaces of functional R-shifts introduced by the author in [13]. The space contains all functions analytic in a ring {h ∈ ℂ : 0 < |h| < ρ}, 0 < ρ ≤ +∞, do not having an essential singularity at the origin. The results obtained include and extend some of those in the author's works [5], [7].
Abstract: In this note we review briefly the subject of bounded and unbounded derivations of operator algebras (C*-and W*-algebras). In particular, we mention some results on closability and continuity of such derivations. This is an invitation to work on three problems in this subject.
Abstract: Several definitions of convolution of ultradistributions of Beurling type as well as of tempered ultradistributions of Beurling type are presented. The definitions are analogous to the known definitions of the convolution of distributions and tempered distributions introduced by C. Chevalley, L. Schwartz, R. Shiraishi, V. S. Vladimirov and others. Similarly to the case of distributions,...
Abstract: In the paper we give sufficient conditions for the existence of convolution of two ultradistributions of Beurling type. We distinguish two types of sufficient conditions: a) conditions in terms of the supports of ultradistributions; b) conditions in terms of subspaces of ultradistributions on which convolution can be defined as a bilinear mapping.
Abstract: 1. Introduction. This paper is the survey of results on the one- and two-sided (continuous) invertibility for some classes of functional operators in Hölder, Lebesgue and Orlicz spaces, which were obtained in the theory of singular integral operators with discrete groups of shifts. In particular, we consider the invertibility in these spaces for binomial and polynomial operators generated...
Abstract: The Kummer solution of the Laguerre differential equationt(d²x/dt²) + (1-t)(dx/dt) - ax = 0, a ∈ ℂ,can be represented by means of the power series$x_{a} = 1 + (at/1!) + (a(a+1)/2!)·(t²/2!) + ... + ((a(a+1)...[(a+n)-1])/n!)·(t^{n}/n!) + ... $Let $x_{a,γ}(t) = e^{γt} x_{a}(t)$. Applying Mikusiński's operational calculus the formulas$d/dt [x_{a}*x_{b}](t) = x_{a+b}(t)$,$(d/dt - γ)[x_{a,γ}*x_{b,γ}](t)...
Abstract: The theory of hyper-Bessel differential operators of arbitrary order m > 1 has been shown to be closely related to the Meijer's G-functions ([9], [10], [18], [20]-[25], [27]). However, most of the operational calculi, integral transforms and solutions to the Bessel type differential equations developed by different authors concern special cases mainly of order m = 2 when the role of...
Abstract: It is shown that from the fact that a homogeneous problem has a unique trivial solution it follows that a non-homogeneous problem has a solution in the Colombeau algebra.
Abstract: The aim of my lecture is to present some results from the theory of generalized analytic functions (GAF for short). I will pay a special attention to deriving from the theory of GAF's a kind of quasi-analyticity principle. Let me say that GAF’s behave in a simple way under basic algebraic and differential operations, and analytic change of variables. As pointed out by B. Ziemian, they form...
Abstract: Boehmians are defined by an algebraic construction which is similar to the construction of a field of quotients. If the construction is applied to a function space and the multiplication is interpreted as convolution, the construction yields a space of generalized functions. Those spaces provide a natural setting for extensions of transforms like the Fourier, Laplace, Radon, or Zak transforms...
Abstract: We give a survey on some recent results for quasidifferentiable functions. Beside some general properties of this type of functions, critical points and the problem of local linearization in a neighborhood of a regular point are discussed. Moreover, the structure of the quasidifferential is investigated. In particular, attention is paid to the problem of finding minimal representatives...
Abstract: A characterization of bounded sets and a boundary value representations in and S'* are given. A decomposition of an ultradistribution, with appropriate assumptions on the corresponding singular spectra, is easily proved.
1. Introduction. In this survey we consider operational calculus as a branch of linear functional analysis. The purpose of this survey is to give some ideas what the modern theory of operational calculus is about. We shall concentrate mainly on the basic problems of the theory, occasionally touching upon analytic and algebraic aspects. In several cases they lead to work of Heaviside and Mikusiński.
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