On Some Properties of the Quaternionic Functional Calculus

In some recent works we have developed a new functional calculus for bounded and unbounded quaternionic operators acting on a quaternionic Banach space. That functional calculus is based on the theory of slice regular functions and on a Cauchy formula which holds for particular domains where the admissible functions have power series expansions. In this paper, we use a new version of the Cauchy formula with slice regular kernel to extend the validity of the quaternionic functional calculus to functions defined on more general domains. Moreover, we show some of the algebraic properties of the quaternionic functional calculus such as the S-spectral radius theorem and the S-spectral mapping theorem. Our functional calculus is also a natural tool to define the semigroup e ^tA when A is a linear quaternionic operator.      

On some notions of convergence for n‐tuples of operators

The aim of this paper is to show that we can extend the notion of convergence in the norm‐resolvent sense to the case of several unbounded noncommuting operators (and to quaternionic operators as a particular case) using the notion of S‐resolvent operator. With this notion, we can define bounded functions of unbounded operators using the S‐functional calculus for n‐tuples of noncommuting operators. The same notion can be extended to the case of the F‐resolvent operator, which is the basis of the F‐functional calculus, a monogenic functional calculus for n‐tuples of commuting operators. We also prove some properties of the F‐functional calculus, which are of independent interest. Copyright © 2013 John Wiley & Sons, Ltd.     

Functional equations for higher logarithms

Following earlier work by Abel and others, Kummer gave in 1840 functional equations for the polylogarithm function Li _m (z) up to m = 5, but no example for larger m was known until recently. We give the first genuine 2-variable functional equation for the 7-logarithm. We investigate and relate identities for the 3-logarithm given by Goncharov and Wojtkowiak and deduce a certain family of functional equations for the 4-logarithm.     

Hypercomplex Functional Calculi and Function Theories in Clifford Analysis

Some properties of hypercomplex functions (the null solutions of the polynomial Dirac operators in Rⁿ⁺¹) in Clifford Analysis are discussed, their hypercomplex functional calculi for an n-tuple non-commuting self-adjoint operators A are constructed by the use of Cauchy integral formulas, the polynomial approaches to functional calculi are also considered. Although these hypercomplex function theories have different representative forms, their hypercomplex functional calculi are the same as the monogenic functional calculus.     

A generalization of a minimax theorem of Fan via a theorem of the alternative

A minimization problem for a functional on a convex subsetC of a normed linear space is considered. Under certain hypotheses, optimality in a certain subset ofC implies the validity of first-order necessary optimality conditions for the problem inC. The result is applied to a problem in optimal periodic control of neutral functional differential equations.This work was partially supported by a grant from Deutsche Forschungsgemeinschaft and by AFOSR under Grant No. AFOSR-84-0398.     

On solutions to the systems of functional Boolean equations

Solutions to the systems of functional Boolean equations are under study. For each of the classes P ₂, T ₀, T ₁, S, T ₀₁, and S ₀₁, the problem is solved of constructing some systems of functional Boolean equations with a given set of functional constants and one functional variable whose unique solution is a given function of the class under consideration. For an arbitrary nonempty set F of n-argument Boolean functions, a system of equations with the functional constants ∨ and & is built with F as the solution set. If F is closed under transition to dual functions then the corresponding system can be constructed without functional constants.     

On the Hyers–Ulam stabilities of functional equations on ‐Banach spaces

The purpose of this article is to generalize the theory of stability of functional equations to the case of n‐Banach spaces. In this article, we prove the generalized Hyers–Ulam stabilities of the Cauchy functional equations, Jensen functional equations and quadratic functional equations on n‐Banach spaces.     

Stabilizers of Functional Menger Systems

A functional Menger system is a set of n-place functions containing n projections and closed under the so-called Menger's composition of n-place functions. We give the abstract characterization for subsets of these functional systems which contain functions having one common fixed point.     

An approximation procedure for solutions to Mayer problems with cost functional not of classC

We consider here control problems in the Mayer form, with a cost functional which is continuous, but not necessarily of classC ¹. The usual necessary conditions for such problems cannot be applied, since they require that the cost functional beC ¹. We describe a convergence procedure, based upon approximations to the cost functional, that will yield an optimal trajectory to such control systems. Criteria are given which justify that this approximation procedure will yield a trajectory and control, satisfying certain prescribed conditions, which minimize the above cost functional within a specified class of such pairs. Examples are presented.     

A note on the stability of Jordan triple higher ring derivations

In the present paper, we establish the stability and the superstability of a functional inequality corresponding to the functional equation fn（xyx） = ∑i＋j＋k=n fi（x）fj （y）fk（x）. In addition, we take account of the problem of Jacobson radical ranges for such functional inequality.     

Nonlinear functional equations satisfied by orthogonal polynomials

Let c be a linear functional defined by its moments c(xⁱ)=c_i for i=0,1,…. We proved that the nonlinear functional equations P(t)=c(P(x)P(αx+t)) and P(t)=c(P(x)P(xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. This equation relates the polynomials of the family with those of the scaled and shifted family. Other types of nonlinear functional equations whose solutions are formal orthogonal polynomials are also presented. Applications to Legendre and Chebyshev polynomials are given. Then, orthogonality with respect to a definite inner product is studied. When c is an integral functional with respect to a weight function, the preceding functional equations are nonlinear integral equations, and these results lead to new characterizations of orthogonal polynomials on the real line, on the unit circle, and, more generally, on an algebraic curve.     

Reflexivity and the sup of linear functionals

A relatively easy proof is given for the known theorem that a Banach space is reflexive if and only if each continuous linear functional attains its sup on the unit ball. This proof simplifies considerably for separable spaces and can be extended to give a proof that a boundedw-closed subsetX of a complete locally convex linear topological space isw-compact if and only if each continuous linear functional attains its sup onX. This work was supported in part by National Science Foundation grant GP-28578.     

The order bidual of <Emphasis Type="Italic">f</Emphasis> -algebras revisited

It is shown that the order bidual X ^~~ of an Archimedean semiprime f -algebra X has a unit element for the Arens multiplication if and only if every positive linear functional on X extends to a positive linear functional on the f -algebra Orth (X) of all orthomorphisms on X.     

The theory of the nonlinear Boltzmann equation for Maxwell molecules in Fourier representation

We study the Cauchy problem for the spatially homogenem Boltzmann equation for true Maxwell molecules. Using the Fourier representation introduced by Bobylev [Bo75],we give a simplified proof of a result proved by Tanaka [Ta78].Moreover, we show by means of simple geometric properties, that Tanaka functional is an entropy decreasing functional for the Boltzmann equation for Maxwell molecules.     

Evaluating inclusion functionals for random convex hulls

Summary The inclusion functional of a random convex set, evaluated at a fixed convex set K, measures the probability that the random convex set contains K. This functional is an analogue of the complement of the distribution function of an ordinary random variable. A methodology is described for evaluating the inclusion functional for the case where the random convex set is generated as the convex hull of n i.i.d. points from a distribution function F in the plane. For general K and F, the inclusion probability is difficult to compute in closed form. The case where K is a straight line segment is examined in detail and, in this situation, a simple answer is given for an interesting class of distributions F.     

Sur la suite de polynômes orthogonaux associée à la formeu=δ c +λ(x−c) −1 L

We show that, ifL is regular, semi-classical functional, thenu is also regular and semi-classical for every complex λ, except for a discrete set of numbers depending onL andc. We give the second order linear differential equation satisfied by each polynomial of the orthogonal sequence associated withu. The cases whereL is either a classical functional (Hermite, Laguerre, Bessel, Jacobi) or a functional associated with generalized Hermite polynomials are treated in detail.

     

Functional law of iterated logarithm for additive functionals of reversible Markov processes

1.IntroductionandMainResultsAssumethat(Xt),.T(T~NorAl)isaPolishspaceE-valuedMarkovprocess,definedon(fi,F,(R),(ot),(P-c)..E),withitssemigroupoftransitionkernels(Pt).Here(ot)isthesemigroupofshiftsonfisuchthatX.(otw)~X. t(w),Vs,tET;(R)isthenaturalfiltration.Throughoutthispaperweassumethat(Pt)issymmetricandergodicwithrespectto(w.r.t.forshort)aprobabilitymeasurepon(E,e)(eistheBorela--fieldofE),i.e.,.Symmetry:(Ptf,g)~(f,Pig):~isfptgdp,acET,if,gCL'(P);.ErgodicitytFOranyfEL'(P),ifPtf~f…     

Positivity of solutions to boundary value problems for infinite functional differential systems

In this paper the problems of unique solvability of boundary value problems for infinite functional differential systems and positivity of their solutions are reduced to analysis of scalar functional equations in the space of measurable essentially bounded functions y:[0,+∞)→(−∞,+∞). Efficient tests of the solvability and the positivity are obtained.     

Variational principles for spectral radii of positive functional operators

Functional operators, i.e., sums of weighted shift operators generated by various maps, are considered. For functional operators with positive coefficients, variational principles for spectral radii are obtained. These principles say that the logarithm of the spectral radius is the Legendre transform of a certain convex functional T defined on the set of probability vector-valued measures and depending on the original dynamical system and the functional space considered. In the subexponential case, we obtain the combinatorial structure of the functional T with the help of the corresponding random walk process constructed according to the dynamical system. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 15, Differential and Functional Differential Equations. Part 1, 2006.