Approximation of harmonic functions on compact sets in ℂ

The direct theorem of the theory of approximation of harmonic functions is established in the case of functions defined on a compact set, the complement of which with respect to is a John domain.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1467–1475, November, 1993.     

On the closure of the sum of two uniform algebras on compact sets in ℂ

Necessary and sufficient conditions on a compact set X in ? and a self-homeomorphism ψ of the plane ? are studied under which any function continuous on X can be approximated uniformly on X by functions of the form p + h ° ψ, where p is a polynomial in a complex variable and h is a rational function whose poles belong to the bounded components of the complement to the compact set ψ(X).     

Compact composition operators between weighted Bergman spaces on convex domains in ℂ n

It is shown that a compact composition operator on a weighted Bergman space over a smoothly bounded strongly convex domain in ⁿ can have no angular derivative. Also, sufficient conditions for the boundedness and the compactness of composition operators defined on Hardy and weighted Bergman spaces are obtained, for situations in which each of the target spaces is enlarged in a natural way.     

Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

In this article we study the (small) Hankel operator h_b on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂⁿ. We completely characterize the Hankel operators h_b that are bounded, compact, and belong to the Schatten ideal S_p, for 0 < p < ∞. In particular, if h_b denotes the Hankel operator on the Hardy space H² (Ω), we prove that h_b is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈e B_p, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H¹ (Ω), a result that is of independent interest.     

Bounded projections and duality on spaces of holomorphic functions in the unit ball ofℂ n

In this paper three Banach spacesA ₀(),A andA ¹() of functions holomorphic in the unit ballB of ⁿ are defined. We exhibit bounded projections fromC ₀(B) ontoA ₀(), fromL ¹(B) ontoA ¹(), and fromL(B) ontoA(). Using these projections, we show thatA ₀()* A ¹() andA ¹()* A().Supported in part by the National Natural Science Foundation of China.     

Improved bounds on weak ε-nets for convex sets

LetS be a set ofn points in ℝ ^d . A setW is aweak ε-net for (convex ranges of)S if, for anyT⊆S containing εn points, the convex hull ofT intersectsW. We show the existence of weak ε-nets of size , whereβ ₂=0,β ₃=1, andβ _d ≈0.149·2 ^d-1(d-1)!, improving a previous bound of Alonet al. Such a net can be computed effectively. We also consider two special cases: whenS is a planar point set in convex position, we prove the existence of a net of sizeO((1/ε) log^1.6(1/ε)). In the case whereS consists of the vertices of a regular polygon, we use an argument from hyperbolic geometry to exhibit an optimal net of sizeO(1/ε), which improves a previous bound of Capoyleas. Work by Bernard Chazelle has been supported by NSF Grant CCR-90-02352 and the Geometry Center. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR-89-21421. Work by Michelangelo Grigni has been supported by NSERC Operating Grants and NSF Grant DMS-9206251. Work by Leonidas Guibas and Micha Sharir has been supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Emo Welzl and Micha Sharir has been supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development. Work by Micha Sharir has also been supported by NSF Grant CCR-91-22103, and by a grant from the Fund for Basic Research administered by the Israeli Academy of Sciences.     

Extension of isometries on the unit sphere of l~p (Γ) space

In this paper,we obtain that every isometry from the unit sphere S(l p (Γ)) of l p (Γ) (1 p ∞,p≠2) onto the unit sphere S(E) of a Banach space E can be extended to be a (real) linear isometry of l p (Γ) onto E,so,we give an affirmative answer to the corresponding Tingley's problem.     

A direct theorem for strictly convex domains in ℂn

For a strictly covex C²-smooth domain Ω??ⁿ and a function f ? Λ^α(Ω) holomorphic in Ω, we construct polynomials p_N, deg p_N<-N, such that $$\left| {f(z) - p_N (z)} \right| \leqslant CN^{ - \alpha } ,z \in \bar \Omega .$$ Bibliography: 12 titles.     

Notes on Schneider’s stability estimates for convex sets

In 2009 Schneider obtained stability estimates in terms of the Banach–Mazur distance for several geometric inequalities for convex bodies in an n-dimensional normed space ${\mathbb{E}^n}$ . A unique feature of his approach is to express fundamental geometric quantities in terms of a single function ${\rho:\mathfrak{B} \times \mathfrak{B} \to \mathbb{R}}$ defined on the family of all convex bodies ${\mathfrak{B}}$ in ${\mathbb{E}^n}$ . In this paper we show that (the logarithm of) the symmetrized ρ gives rise to a pseudo-metric d _D on ${\mathfrak{B}}$ inducing, from our point of view, a finer topology than Banach–Mazur’s d _BM . Further, d _D induces a metric on the quotient ${\mathfrak{B}/{\rm Dil}^+}$ of ${\mathfrak{B}}$ by the relation of positive dilatation (homothety). Unlike its compact Banach–Mazur counterpart, d _D is only “boundedly compact,” in particular, complete and locally compact. The general linear group ${{\rm GL}(\mathbb{E}^n)}$ acts on ${\mathfrak{B}/{\rm Dil}^+}$ by isometries with respect to d _D , and the orbit space is naturally identified with the Banach–Mazur compactum ${\mathfrak{B}/{\rm Aff}}$ via the natural projection ${\pi:\mathfrak{B}/{\rm Dil}^+\to\mathfrak{B}/{\rm Aff}}$ , where Aff is the affine group of ${\mathbb{E}^n}$ . The metric d _D has the advantage that many geometric quantities are explicitly computable. We show that d _D provides a simpler and more fitting environment for the study of stability; in particular, all the estimates of Schneider turn out to be valid with d _BM replaced by d _D .     

A Petrov–Galerkin RBF method for diffusion equation on the unit sphere

This paper concerns a numerical solution for the diffusion equation on the unit sphere. The given method is based on the spherical basis function approximation and the Petrov–Galerkin test discretization. The method is meshless because spherical triangulation is not required neither for approximation nor for numerical integration. This feature is achieved through the spherical basis function approximation and the use of local weak forms instead of a global variational formulation. The local Petrov–Galerkin formulation allows to compute the integrals on small independent spherical caps without any dependence on a connected background mesh. Experimental results show the accuracy and the efficiency of the new method.     

Estimating non-convex production sets – imposing convex input sets and output sets in data envelopment analysis

In this paper, we present Data Envelopment Analysis models for estimating non-convex production sets. In contrast to the Banker et al. (Banker, R.D., Charnes, A. and Cooper, W.W., 1984. Management Science 30 (9), 1078–1092) model, these models give statistically consistent estimators for production sets that are non-convex, but do have convex input sets and/or convex output sets, as is typically assumed in micro-economic theory. In addition, these models suffer less from a finite sample error than the Free Disposable Hull (Deprins, D., Simar, L., Tulkens, H., 1984. In: Marchand, M., Pestieu, P., Tulkens, H. (Eds.), The Performance of Public Enterprises. North-Holland, Amsterdam, pp. 243–267) model.     

The maximum number of ways to stabn convex nonintersecting sets in the plane is 2n−2

LetS be a collection ofn convex, closed, and pairwise nonintersecting sets in the Euclidean plane labeled from 1 ton. A pair of permutations $$\left\{ {(i_1 ,i_2 ,...,i_{n - 1} ,i_n ,),(i_n ,i_{n - 1} ,...,i_2 ,i_1 ,)} \right\}$$ is called ageometric permutation of S if there is a line that intersects all sets ofS in this order. We prove thatS can realize at most 2n?2 geometric permutations. This upper bound is tight.     

Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of ℂ n

We give a Schwarz-Pick estimate for bounded holomorphic functions on unit ball in Cn, and generalize some early work of Schwarz-Pick estimates for bounded holomorphic functions on unit disk in C.     

Convergence of the method of fundamental solutions for Poisson’s equation on the unit sphere

Solutions of boundary value problems of the Laplace equation on the unit sphere are constructed by using the fundamental solution

Zonal spherical function on the Lie ball in ℂn

In this paper, we give the explicit formula for the zonal spherical function on the Lie ball in ?ⁿ.     

Eversive maps of bounded convex domains in ℂ n+1

A duality principle, relating the geometry of the Kobayashi metric with the CR geometry of the boundaries of smoothly bounded, strongly convex domains in ℂ ⁿ⁺¹ is established. A characterization of the holomorphic Jacobi vector fields of those domains is also given.