Traces of pluriharmonic functions on curves

We prove that, if γ is a simple smooth curve in the unit sphere inC ⁿ, the space o pluriharmonic functions in the unit ball, continuous up to the boundary, has a trace of finite cof dimension in the space of all continuous functions on the curve. First author partially supported by the Swedish Natural Science Research Council. Second author partially supported by CICYT grant PB85-0374.     

Optimal control of nonlinear hereditary systems

A function space Λ is introduced for the study of nonlinear hereditary differential equations. The properties of Λ include: it is a Banach space under the supremum norm, the continuous functions constitute a closed proper subspace, and the unit ball is sequentially compact in the weak-¹ topology. Existence, uniqueness, and continuous dependence results are obtained for solutions of a broad class of initial value problems. An optimization problem is formulated for systems which are affine in the control, and solutions are approximated by means of a sequence of problems which are finite-dimensional in the control.     

A Remark on the Local Lipschitz Continuity of Vector Hysteresis Operators

It is known that the vector stop operator with a convex closed characteristic Z of class C ¹ is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping n is Lipschitz continuous on the boundary Z of Z. We prove that in the regular case, this condition is also necessary.     

BEST MONOTONE L_■-APPROXIMATIONS IN SEVERAL VARIABLES

Given a continuous function f defined on the unit cube of R~n and a convexfunction _t,_t(0)-0,_t(x)>0,for x>0,we prove that the set ofbest L~(t)-approximations by monotone functions has exactly one elementft,which is also a continuous function.Moreover if the family of convexfunctions {_t}t>0 converges uniformly on compact sets to a function _0,then the best approximation f_t→f_0 uniformly,as t→0,where fo is thebest approximation of f within the Orlicz space L~(0) The best approxima-tions{f_t}are obtained as well as minimizing integrals or the Luxemburgnorm     

A spline collocation method for singular integral equations with piecewise continuous coefficients

We consider the collocation method with piecewise linear trial functions for systems of singular integral equations with Cauchy kernel and piecewise continuous coefficients. Necessary and sufficient conditions for the stability in L² are given. The results are obtained in the case of a closed Ljapunov curve as well as in the case of an interval. The proof of the main theorem is based on a modification of the Banach algebra technique established in the local principle by Gohberg and Krupnik [2]. Our results extend those obtained by Prößdorf and Schmidt [9, 10] from the case of continuous coefficients and unit circle to the case of piecewise continuous coefficients.     

Regularity of Skorohod integral processes based on integrands in a finite Wiener chaos

Summary Letf be a square integrable kernel on them-dimensional unit cube,U the Skorohod integral process in them ^th Wiener chaos associated with it. Isoperimetric inequalities for functions on Wiener space yield the exponential integrability of the increments ofU. To this result we apply the majorizing measure technique to show thatU possesses a continuous version and give an upper bound of its modulus of continuity.     

Fractional integrals of periodic functions of several variables

In this paper it has been systematically studied the imbedding properties of fractional integral operators of periodic functions of several variables, and isomorphic properties of fractional integral operators in the spaces of Lipschitz continuous functions. It has also been proved that the space of fractional integration, the space of Lipschitz continuous functions and the Sobolev space are identical in L²-norm. Results obtained here are not true for fractional integrals (or Riesz potentials) in ℝ ⁿ . Supported by NNSFC     

Characterization and properties of extreme operators intoC(Y)

LetE be a real (or complex) Banach space,Y a compact Hausdorff space, andC(Y) the space of real (or complex) valued continuous functions onY. IfT is an extreme point in the unit ball of bounded linear operators fromE intoC(Y), then it is shown thatT ^* maps (the natural imbedding inC(Y) ^* of)Y into the weak ^*-closure of extS(E ^*), provided thatY is extremally disconnected, orE=C(X), whereX is a dispersed compact Hausdorff space.     

The capability of approximation for neural networks interpolant on the sphere

Compared with planar hyperplane, fitting data on the sphere has been an important and active issue in geoscience, metrology, brain imaging, and so on. In this paper, using a functional approach, we rigorously prove that for given distinct samples on the unit sphere there exists a feed‐forward neural network with single hidden layer which can interpolate the samples, and simultaneously near best approximate the target function in continuous function space. Also, by using the relation between spherical positive definite radial basis functions and the basis function on the Euclidean space ?^{d + 1}, a similar result in a spherical Sobolev space is established. Copyright © 2010 John Wiley & Sons, Ltd.     

Representable Functions on the Unit Ball of a Banach Space

In this paper we treat the problem of integral representation of analytic functions over the unit ball of a complex Banach space X using the theory of abstract Wiener spaces. We define the class of representable functions on the unit ball of X and prove that this set of functions is related with the classes of integral k–homogeneous polynomials, integral holomorphic functions and also with the set of L ^p –representable functions on a Banach space.     

Isomorphisms of lattices of subalgebras of semirings of continuous nonnegative functions

In this work, lattice isomorphisms of semirings C ⁺(X) of continuous nonnegative functions over an arbitrary topological space X are characterized. It is proved that any isomorphism of lattices of all subalgebras with a unit of semirings C ⁺(X) and C ⁺(Y) is induced by a unique isomorphism of semirings. The same result is also correct for lattices of all subalgebras excepting the case of two-point Tychonovization of spaces.     

A NOTE ON WEAKLY CONTINUOUS BANACH SPACE VALUED FUNCTIONS

Abstract

A simple remark on the localization of the extreme points of the unit ball of the dual of the space of weakly continuous functions or weak* continuous functions give some new insight on these spaces and simplifies proofs in [ADLR 92] and [LO 91].     

Note on a Result of Kerman and Weit II

Let G be a locally compact topological group, equipped with a fixed left Haar measure μ. We show that if f is a compactly supported real valued continuous function on G which has a unique maximum or a unique minimum at a point in G, then the space generated by the span of left translations of {f ⁿ ∣n=1,2,3,…} is dense in L ^p (G,μ), 1≤p<∞, in the space of continuous functions, continuous compactly supported functions and in the space of continuous functions vanishing at ∞. Similar results are true when the group G is substituted by G-spaces with compact isotropy group.     

Discontinuity of Best Harmonic Approximants

LetDR²be the open unit disk. We consider best harmonic approximation to functions continuous onD. In a basic paper, Haymanet al.characterized best harmonic approximants which are themselves continuous onD. In this paper we give sufficient conditions and many simple examples of functions continuous onDwhich have no best harmonic approximants which are continuous onD.     

Pre-compactness of isospectral sets for the neumann operator on planar domains

The Neumann operator maps the boundary value of a harmonic function tc its normal derivative. The inverse spectral properties of the Neumann operator associated to smooth, planar, Jordan curves are studied. The Riemann mapping theorem is used tc parametrize the set of planar Jordan curves by positive functions on the unit circle. By studying the zeta function associated to the spectrum, it is shown that isospectral sets of these functions are pre-compact in the topology of the L²-Sobolev space of order 5/2 - [euro]. Spectral criteria are given for the limiting curves of an isospectral set to be Jordan. A spectrally determined lower bound on the area of the interior of the curve is given.     

Curves and Surfaces Construction Based on New Basis with Exponential Functions

By incorporating two exponential functions into the cubic Bernstein basis functions, a new class of λμ-Bernstein basis functions is constructed. Based on these λμ-Bernstein basis functions, a kind of λμ-Bézier-like curve with two shape parameters, which include the cubic Bernstein-Bézier curve, is proposed. The C ¹ and C ² continuous conditions for joining two λμ-Bézier-like curves are given. By using tensor product method, a class of rectangular Bézier-like patches with four shape parameters is shown. The G ¹ and G ² continuous conditions for joining two rectangular Bézier-like patches are derived. By incorporating three exponential functions into the cubic Bernstein basis functions over triangular domain, a new class of λμη-Bernstein basis functions over triangular domain is also constructed. Based on the λμη-Bernstein basis functions, a kind of triangular λμη-Bézier-like patch with three shape parameters, which include the triangular Bernstein-Bézier cubic patch, is presented. The conditions for G ¹ continuous smooth joining two triangular λμη-Bézier-like patches are discussed. The shape parameters serve as tension parameters and have a predictable adjusting role on the curves and patches.     

Selezioni continue o misurabili

We give a characterization of set-valued mappings from a topological (measure) space into the class of real non-empty intervals admitting a continuous (measurable) selection. As an application we obtain a characterization of set-valued functions defined on IR ⁿ admitting an approximately continuous or an approximately continuous and almost everywhere continuous selection.     

Continuous embeddings in harmonic mixed norm spaces on the unit ball in ? n

In this paper continuous embeddings in spaces of harmonic functions with mixed norm on the unit ball in ? ⁿ are established, generalizing some Hardy-Littlewood embeddings for similar spaces of holomorphic functions in the unit disc. Differences in indices between the spaces of harmonic and holomorphic spaces are revealed. As a consequence an analogue of classical Fejér-Riesz inequality is obtained. Embeddings in the special case of Riesz systems are also established.