Dirichlet problem for pluriholomorphic functions of two complex variables

In this paper the Dirichlet problem for pluriholomorphic functions of two complex variables is investigated. The key point is the relation between pluriholomorphic functions and pluriharmonic functions. The link is constituted by the Fueter-regular functions of one quaternionic variable. Previous results about the boundary values of pluriharmonic functions and new results on L² traces of regular functions are applied to obtain a characterization of the traces of pluriholomorphic functions.     

Rota–Baxter algebras and left weak composition quasi-symmetric functions

Motivated by a question of Rota, this paper studies the relationship between Rota–Baxter algebras and symmetric-related functions. The starting point is the fact that the space of quasi-symmetric functions is spanned by monomial quasi-symmetric functions which are indexed by compositions. When composition is replaced by left weak composition (LWC), we obtain the concept of LWC monomial quasi-symmetric functions and the resulting space of LWC quasi-symmetric functions. In line with the question of Rota, the latter is shown to be isomorphic to the free commutative nonunitary Rota–Baxter algebra on one generator. The combinatorial interpretation of quasi-symmetric functions by P-partitions from compositions is extended to the context of left weak compositions, leading to the concept of LWC fundamental quasi-symmetric functions. The transformation formulas for LWC monomial and LWC fundamental quasi-symmetric functions are obtained, generalizing the corresponding results for quasi-symmetric functions. Extending the close relationship between quasi-symmetric functions and multiple zeta values, weighted multiple zeta values, and a q-analog of multiple zeta values are investigated, and a decomposition formula is established.     

r-convex functions

A family of real functions, calledr-convex functions, which represents a generalization of the notion of convexity is introduced. This family properly includes the family of convex functions and is included in the family of quasiconvex functions. Some properties ofr-convex functions are derived and relations with other generalizations of convex functions are discussed.Portions of this paper were presented at the 7th Mathematical Programming Symposium 1970, The Hague, The Netherlands.Research for this paper was supported in part by the Gerard Swope Fund at the Technion.     

关于α-凸函数的一个子类的Fekete-Szeg(o)不等式

摘要：引入α-凸函数的一个子类M（α;A,B）,讨论了这个函数类上的Fekete-Szego不等式,得到了准确的结果,推广了一些作者的结果．最后,给出了不等式在利用Hadamard积定义的函数类上的两个应用．     

Approximation by translates of refinable functions

Summary. The functions are refinable if they are combinations of the rescaled and translated functions . This is very common in scientific computing on a regular mesh. The space of approximating functions with meshwidth is a subspace of with meshwidth . These refinable spaces have refinable basis functions. The accuracy of the computations depends on , the order of approximation, which is determined by the degree of polynomials that lie in . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions are known only through the coefficients in the refinement equation – scalars in the traditional case, matrices for multiwavelets. The scalar "sum rules" that determine are well known. We find the conditions on the matrices that yield approximation of order from . These are equivalent to the Strang–Fix conditions on the Fourier transforms , but for refinable functions they can be explicitly verified from the . Received August 31, 1994 / Revised version received May 2, 1995     

Functions of a quaternion variable which are gradients of real-valued functions

We consider the collection of functions of one quaternion variable which can be expressed asG(Y) whereY is a real-valued quaternion function andG is a differential operator which corresponds to the gradient of real variable theory. Integral theorems for such functions are given, together with necessary and sufficient conditions for a function to be a gradient function, in terms of its Frechet derivative. The extended complex analytic functions, the Fueter functions, and the momentum-energy density functions are seen to be gradient functions which correspond to biharmonic, harmonic, and wave functions respectively.     

When separability of the space of ultra-extremal functions is preserved

Abstract

The ultrametrically injective hull T_X of an ultrametric space (X, d) is investigated by viewing it as the space of ultra-extremal functions over X. It turns out that the ultra-extremal functions are also ultra-Ka?etov functions, satisfying two inequalities derived from the strong triangle inequality. We shall compare the ultra-extremal functions with some classes of functions defined with the help of one of the two inequalities from the definition of ultra-Kat?tov functions. We shall consider the question of when separability of the space of ultra-extremal functions is preserved.     

Rings of continuous functions. Algebraic aspects

This survey paper is devoted to the algebraic theory of rings of continuous functions. The author considers the relationship between topological spaces and the corresponding rings of continuous functions. The ring properties of rings of continuous functions are discussed. The general theory of sheaves of rings is considered briefly. The importance of rings of functions in mathematics is demonstrated.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 29, pp. 119–191, 1991.     

A class of uniform functions and its relationship with the class of measurable functions

Borel, Lebesgue, and Hausdorff described all uniformly closed families of real-valued functions on a set T whose algebraic properties are just like those of the set of all continuous functions with respect to some open topology on T. These families turn out to be exactly the families of all functions measurable with respect to some σ-additive and multiplicative ensembles on T. The problem of describing all uniformly closed families of bounded functions whose algebraic properties are just like those of the set of all continuous bounded functions remained unsolved. In the paper, a solution of this problem is given with the help of a new class of functions that are uniform with respect to some multiplicative families of finite coverings on T. It is proved that the class of uniform functions differs from the class of measurable functions.     

Global optimization of Hölder functions

We propose a branch-and-bound framework for the global optimization of unconstrained Hölder functions. The general framework is used to derive two algorithms. The first one is a generalization of Piyavskii's algorithm for univariate Lipschitz functions. The second algorithm, using a piecewise constant upper-bounding function, is designed for multivariate Hölder functions. A proof of convergence is provided for both algorithms. Computational experience is reported on several test functions from the literature.     

Harmonic functions and exit boundary of superdiffusion

All positive harmonic functions in an arbitrary domain of a Euclidean space can be described in terms of the so-called exit boundary. This was established in 1941 by R.S. Martin. A probabilistic approach to the Martin theory is due to Doob and Hunt. It was extended later to harmonic functions associated with a wide class of Markov processes. The subject of this paper are harmonic functions associated with a superdiffusion X (we call them X-harmonic). The results of Evans and Perkins imply the existence and uniqueness of an integral representation of positive X-harmonic functions through extreme functions. An outstanding problem is to find all extremal functions (they are in 1-1 correspondence with the points of the exit boundary).An interest in X-harmonic functions is motivated, in part, by the fact that each of them provides a way of conditioning a superprocess. Path properties of conditioned superprocesses (corresponding to various special X-harmonic functions) were investigated by a number of authors.Important classes of X-harmonic functions are related to positive solutions of semilinear partial differential equations. Almost nothing is known about their decomposition into extreme elements. A progress in this direction may create new tools for investigating solutions of the equations.The goal of this paper is to summarize all known facts about X-harmonic functions, to present the results of various authors in a more general setting in a unified form and to outline a program of further work.     

DISTRIBUTION OF MULTIPLICATIVE FUNCTIONS DEFINED ON SEMIGROUPS

Abstract

The value distribution problem for real-valued multiplicative functions defined on an additive arithmetical semigroup is examined. We prove that, in contrast to the classical theory of number-theoretic functions defined on the semigroup of natural numbers, this problem is equivalent to that for additive functions only under some extra condition. In this way, applying the known results for additive functions we derive general sufficient conditions for the existence of a limit law for appropriately normalized multiplicative functions.     

Some Extensions of Loewner's Theory of Monotone Operator Functions

Several extensions of Loewner's theory of monotone operator functions are given. These include a theorem on boundary interpolation for matrix-valued functions in the generalized Nevanlinna class. The theory of monotone operator functions is generalized from scalar-to matrix-valued functions of an operator argument. A notion of κ-monotonicity is introduced and characterized in terms of classical Nevanlinna functions with removable singularities on a real interval. Corresponding results for Stieltjes functions are presented.     

The greatest regular subalgebra of certain Banach algebras of vector-valued functions

Given an arbitrary commutative complex Banach algebraA, it is shown that, for various classical Banach algebras ofA-valued functions, the greatest regular subalgebra consists precisely of those functions which map into the greatest regular subalgebra ofA. The main result covers the case of continuous and differentiable functions, Lipschitz functions, and Bochner integrable functions on a locally compact abelian group. The principal tools are from the theory of tensor products of Banach algebras.     

Godambe estimating functions and asymptotic optimal inference

Godambe (1985) introduced a class of optimum estimating functions which can be regarded as a generalization of quasilikelihood score functions. The “optimality” established by Godambe (1985) within a certain class is for estimating functions and it is based on finite samples. The question that arises naturally is what (if any) asymptotic optimality properties do the estimators and tests based on optimum estimating functions possess. In this paper, we establish, via presenting a convolution theorem, asymptotic optimality of estimators and tests obtained from Godambe optimum estimating functions. It is noted that we do not require the knowledge of the likelihood function.     

On the ranks of bent functions

The rank of a bent function is the 2-rank of the associated symmetric 2-design. In this paper, it is shown that it is an invariant under the equivalence relation among bent functions. Some upper and lower bounds of ranks of general bent functions, Maiorana–McFarland bent functions and Desarguesian partial spread bent functions are given. As a consequence, it is proved that almost every Desarguesian partial spread bent function is not equivalent to any Maiorana–McFarland bent function.     

Some new inequalities in geometry and analysis

The paper announces some new inequalities that refer to broken lines, curves and real and complex functions. Their derivation is based on a new principle of angles and lengths for curves. The inequality in the complex analysis, called the principle of derivatives, is valid for analytic functions in arbitrary domains and extends to a broad class of sufficiently smooth complex functions. A new inequality follows for real functions of two variables concerning their level sets. For all cases mentioned above, both for curves and functions, we obtain some analogues of the second fundamental theorem in Nevanlinna theory of meromorphic functions. At the end we discuss a new point-domain inequality dealing with finite point sets in an arbitrary domain.     

Construction of orthonormal wavelets using Kampé de Fériet functions

One of the main aims of this paper is to bridge the gap between two branches of mathematics, special functions and wavelets. This is done by showing how special functions can be used to construct orthonormal wavelet bases in a multiresolution analysis setting. The construction uses hypergeometric functions of one and two variables and a generalization of the latter, known as Kampé de Fériet functions. The mother wavelets constructed by this process are entire functions given by rapidly converging power series that allow easy and fast numerical evaluation. Explicit representation of wavelets facilitates, among other things, the study of the analytic properties of wavelets.

     

Expansion over a system of shifts of pyramidal functions

A method for approximation of functions of two variables by a linear combination of nonnegative piecewise linear functions with a compact support is presented. The crucial idea of this method consists in an “integral” calculation method for the coefficients. The accuracy of the approximation in the spaces of continuous and integrable functions is proved to have the same order as the best approximation by piecewise linear functions.     

On the outcome of false images of linear <Emphasis Type="Italic">k</Emphasis>-valued functions for composite numbers <Emphasis Type="Italic">k</Emphasis> when the number of variables increases

The problem of constructing discrete functions such that parts of their value sets determine (generate) arbitrary linear functions is considered. A case in which k is a prime number was considered earlier by the author. It is proved that the existence of such partial functions wshen the number of independent variables is no less then two implies they exists for any arbitrary greater number of independent variables. Upper estimates linear with respect to the number of independent variables are proved for the size of the domain of universal functions. The existence of two-variable universal functions is proved for sufficiently large k.