On generalized means and generalized convex functions

Properties of generalized convex functions, defined in terms of the generalized means introduced by Hardy, Littlewood, and Polya, are easily obtained by showing that generalized means and generalized convex functions are in fact ordinary arithmetic means and ordinary convex functions, respectively, defined on linear spaces with suitably chosen operations of addition and multiplication. The results are applied to some problems in statistical decision theory.This research was supported by Project No. NR-047-021, Contract No. N00014-75-C-0569 with the Center for Cybernetic Studies, The University of Texas, Austin, Texas, and by NSF Grant No. ENG-76-10260 at Northwestern University, Evanston, Illinois.     

Hermite functions and weighted spaces of generalized functions

We prove that the Hermite functions are an absolute Schauder basis for many weighted spaces of (ultra)differentiable functions and ultradistributions including the space of Fourier hyperfunctions. The coefficient spaces are also determined. Dedicated to Professor H.-G. Tillmann on the occasion of his 80th birthday     

A generalized Laplace transform of generalized functions

- . . . , .     

Harmonic generalized functions in generalized function algebras

We investigate basic properties of harmonic generalized functions within the framework of J. F. Colombeau??s theory of generalized functions. In particular, we present various theorems concerning the Maximum principle, Liouville??s theorem, singularities and Poisson formula.     

Real analytic generalized functions

Real analytic generalized functions are introduced and investigated. The analytic singular support and analytic wave front of a generalized function in are introduced and described. Authors’ addresses: S. Pilipović, Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovića 4, 21000 Novi Sad, Serbia; D. Scarpalezos, U.F.R. de Mathématiques, Université Paris 7, 2 Place Jussieu, Paris 5^e, 75005, France; V. Valmorin, Université des Antilles et de la Guyane, Département Math-Info, Campus de Fouillole, 97159 Pointe á Pitre Cedex, France     

Real analytic generalized functions

Real analytic generalized functions are introduced and investigated. The analytic singular support and analytic wave front of a generalized function in G(W){\cal G}(\Omega) are introduced and described.     

On generalized Christoffel functions

The generalized Christoffel function λ _p,q,n (dμ;x) (0<p<∞, 0≦q<∞) with respect to a measure dμ on R is defined by

$\lambda_{p,q,n}(d\mu;x)=\inf_{Q\in\mathbf{P}_{n-1},\ Q(x)=1}\int_{\mathbf{R}} \big|Q(t)\big|^p {|t-x|}^q\, d\mu(t).$

The novelty of our definition is that it contains the factor |t?x| ^q , which is of particular interest. Its properties are discussed and estimates are given. In particular, upper and lower bounds for generalized Christoffel functions with respect to generalized Jacobi weights are also provided.

     

Homotopy analysis method for generalized Benjamin–Bona–lMahony equation

An analytic technique, the homotopy analysis method (HAM), is applied to solve the generalized Benjamin–Bona–Mahony (BBM) equation. An explicit series solution is given, different from traditional analytic techniques, our approach is independent of knowing some parameters. This analytic method provides us with a new way to obtain series solutions of such problems. The homotopy analysis method contains the auxiliary parameter ħ, which provides us with a simple way to adjust and control the convergence region of series solution.     

Tempered generalized functions and Hermite expansions

In this work we introduce an algebra of tempered generalized functions. The tempered distributions are embedded in this algebra via their Hermite expansions. The Fourier transform is naturally extended to this algebra in such a way that the usual relations involving multiplication, convolution and differentiation are valid. Furthermore, we give a generalized Itô formula in this context and some applications to stochastic analysis.     

Bent and generalized bent Boolean functions

In this paper, we investigate the properties of generalized bent functions defined on ${\mathbb{Z}_2^n}$ with values in ${\mathbb{Z}_q}$ , where q ≥ 2 is any positive integer. We characterize the class of generalized bent functions symmetric with respect to two variables, provide analogues of Maiorana–McFarland type bent functions and Dillon’s functions in the generalized set up. A class of bent functions called generalized spreads is introduced and we show that it contains all Dillon type generalized bent functions and Maiorana–McFarland type generalized bent functions. Thus, unification of two different types of generalized bent functions is achieved. The crosscorrelation spectrum of generalized Dillon type bent functions is also characterized. We further characterize generalized bent Boolean functions defined on ${\mathbb{Z}_2^n}$ with values in ${\mathbb{Z}_4}$ and ${\mathbb{Z}_8}$ . Moreover, we propose several constructions of such generalized bent functions for both n even and n odd.     

Generalized martingales,generalized Markov chains and generalized harmonic functions

Summary This paper introduces and studies a generalization of the notion of martingale which allows for a generalization of the concept of a Markov chain and a generalization of the concept of harmonic and superharmonic functions. The theory is supported by examples and techniques that suggest the natural character of the material developed.Deceased. Please address correspondence on Prof. Magda Peligrad; Department of Mathematical Sciences, University of Cincinnati, Mail Location 25, Cincinnati, OH 45221 USA     

Generating functions and generalized alternating subsets

The notion of an alternating subset of integers is generalized to the notion of an (α, β)-alternating subset of integers, where α and β are arbitrary positive integers. The problem of enumerating the (α, 1)-alternating k-subsets of {1 2, …, n} is solved by means of a generating function approach related to the multisections of series.     

Wiener-Hopf operators and generalized analytic functions

An almost periodic generalization of H+C is defined and analyzed. Applications of this analysis are made to the type II index theory of Wiener-Hopf operators with almost periodic symbols.Supported in part by the National Science Foundation.     

Generalized convex functions and generalized differentials

We study some classes of generalized convex functions, using a generalized differential approach. By this we mean a set-valued mapping which stands either for a derivative, a subdifferential or a pseudo-differential in the sense of Jeyakumar and Luc. Such a general framework allows us to avoid technical assumptions related to specific constructions. We establish some links between the corresponding classes of pseudoconvex, quasiconvex and another class of generalized convex functions we introduced. We devise some optimality conditions for constrained optimization problems. In particular, we get Lagrange–Kuhn–Tucker multipliers for mathematical programming problems.