Global maxima of real-valued functions

In this paper, sufficient conditions for a local maximum to be global are discussed. The result is that a real continuously differentiable functionf, defined on a subset ofR ⁿ, under fairly weak conditions, is unimodal iff has a strict local maximum at any stationary point.The author would like to thank the referee for detailed and constructive comments.     

Inversion of real-valued functions and applications

This work is devoted to a systematic study of the inversion of nondecreasing one variable extended real-valued functions. Its results are preparatory for a new duality theory for quasiconvex problem [6]. However the question arises in a variety of situations and as such deserves a separate treatment. Applications to topology, probability theory, monotone rearrangements, convex analysis are either pointed out or sketched.

---

Zusammenfassung In dieser Arbeit wird systematisch die Umkehrung monoton nichtfallender Funktionenf: {–, +} studiert. Die Ergebnisse bilden die Grundlage für eine neue Dualitätstheorie quasikonvexer Probleme [6]. Da jedoch die Fragestellung bei einer ganzen Anzahl weiterer Situationen auftritt, verdient sie eine gesonderte Behandlung. Anwendungen in der Topologie, Wahrscheinlichkeitstheorie, monotonen Umordnungen und in der konvexen Analysis werden aufgezeigt und skizziert.

     

Completely normal frames and real-valued functions

Up to now point-free insertion results have been obtained only for semicontinuous real functions. Notably, there is now available a setting for dealing with arbitrary, not necessarily (semi-)continuous, point-free real functions, due to Gutiérrez García, Kubiak and Picado, that gives point-free topology the freedom to deal with general real functions only available before to point-set topology. As a first example of the usefulness of that setting, we apply it to characterize completely normal frames in terms of an insertion result for general real functions. This characterization extends a well-known classical result of T. Kubiak about completely normal spaces. In addition, characterizations of completely normal frames that extend results of H. Simmons for topological spaces are presented. In particular, it follows that complete normality is a lattice-invariant property of spaces, correcting an erroneous conclusion in [Y.-M. Wong, Lattice-invariant properties of topological spaces, Proc. Amer. Math. Soc. 26 (1970) 206-208].     

Second-order subdifferentials and convexity of real-valued functions

In this paper we show that the positive semi-definiteness (PSD) of the Fréchet and/or Mordukhovich second-order subdifferentials can recognize the convexity of C¹ functions. However, the PSD is insufficient for ensuring the convexity of a locally Lipschitz function in general. A complete characterization of strong convexity via the second-order subdifferentials is also given.     

Feebly compact paratopological groups and real-valued functions

We present several examples of feebly compact Hausdorff paratopological groups (i.e., groups with continuous multiplication) which provide answers to a number of questions posed in the literature. It turns out that a 2-pseudocompact, feebly compact Hausdorff paratopological group $G$ can fail to be a topological group. Our group $G$ has the Baire property, is Fréchet–Urysohn, but it is not precompact. It is well known that every infinite pseudocompact topological group contains a countable non-closed subset. We construct an infinite feebly compact Hausdorff paratopological group $G$ all countable subsets of which are closed. Another peculiarity of the group $G$ is that it contains a nonempty open subsemigroup $C$ such that $C^{-1}$ is closed and discrete, i.e., the inversion in $G$ is extremely discontinuous. We also prove that for every continuous real-valued function $g$ on a feebly compact paratopological group $G$ , one can find a continuous homomorphism $\varphi $ of $G$ onto a second countable Hausdorff topological group $H$ and a continuous real-valued function $h$ on $H$ such that $g=h\circ \varphi $ . In particular, every feebly compact paratopological group is $\mathbb{R }_3$ -factorizable. This generalizes a theorem of Comfort and Ross established in 1966 for real-valued functions on pseudocompact topological groups.     

Abstract convexity of extended real-valued ICR functions

The theory of non-negative increasing and co-radiant (ICR) functions defined on ordered topological vector spaces has been well developed. In this article, we present the theory of extended real-valued ICR functions defined on an ordered topological vector space X. We first give a characterization for non-positive ICR functions and examine abstract convexity of this class of functions. We also investigate polar function and subdifferential of these functions. Finally, we characterize abstract convexity, support set and subdifferential of extended real-valued ICR functions.     

Algebras of real-valued continuous functions in product spaces

Let {X_i:iϵI} be an arbitrary family of spaces, we say that the cartesian product X has the approximation property when C(X) coincides with the Algebra on X generated by the functions which depend on one variable. In this paper we study the problem of characterizing topologically when an arbitrary product space has the approximation property. We prove that if X is an uncountable pseudo-ℵ₁-compact P-space, then X×Y has the approximation property if, and only if, X×Y is pseudo-ℵ₁-compact. As a corollary we obtain the following characterization for P-spaces: Let X and Y be P-spaces, then X×Y has the approximation property if, and only if, X or Y is countable or X×Y is pseudo-ℵ₁-compact.     

Porosity and typical properties of real-valued continuous functions

Some well-known theorems on typical properties of real-valued continuous functions defined on [0, 1] are improved using the notion of porosity.     

Some properties of algebras of real-valued measurable functions

Measures and measurable functions are used primarily as tools for carrying out various calculations to increase our knowledge. We learn how to combine them in various ways by studying real analysis; a very useful subject on which very much has been written. In this paper, we regard measurable functions as algebras of real-valued functions (or equivalence classes of them) on a set or topological space under point-wise addition, multiplication, or lattice operations and our techniques resemble closely those used to study algebras of continuous functions. This is done by examining a number of explicit examples including Borel and Lebesgue measures and measurable functions.     

Integral transforms of generalized functions

One surveys the investigations on the integral transforms of generalized functions. Basically, the survey convers the papers and monographs published after 1974.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 20, pp. 78–115, 1982.     

Generating functions via integral transforms

In this paper, we use some integral transforms to derive, for a polynomial sequence {Pn(x)}_n?0, generating functions of the type , starting from a generating function of type , where {γn}_n?0 is a real numbers sequence independent on x and t. That allows us to unify the treatment of a generating function problem for many well-known polynomial sequences in the literature.     

Injective spaces of real-valued functions with the baire property

Generalizing the technique used by S.A. Argyros in [3], we give a lemma from which certain Banach spaces are shown to be non-injective. This is applied mainly to study the injectivity of spaces of real-valued Borel functions and functions with the Baire property on a topological space. The results obtained in this way do not follow from previous works about this matter.     

Estimates for the Fourier-Bessel transforms of multivariate functions

Two estimates useful in applications are proved for the Fourier-Bessel (or Hankel) transform in the space $\mathbb{L}_2 \left( {\mathbb{R}_ + ^2 } \right)$ for some classes of two-variable functions characterized by a generalized modulus of continuity.