Jordan Structures in Harmonic Functions and Fourier Algebras on Homogeneous Spaces

We study the Jordan structures and geometry of bounded matrix-valued harmonic functions on a homogeneous space and their analogue, the harmonic functionals, in the setting of Fourier algebras of homogeneous spaces.Supported by EPSRC grant GR/G91182 and NSERC grant 7679.     

A polynomial of graphs on surfaces

ribbon graphs , i.e., graphs realized as disks (vertices) joined together by strips (edges) glued to their boundaries, corresponding to neighbourhoods of graphs embedded into surfaces. We construct a four-variable polynomial invariant of these objects, the ribbon graph polynomial, which has all the main properties of the Tutte polynomial. Although the ribbon graph polynomial extends the Tutte polynomial, its definition is very different, and it depends on the topological structure in an essential way. Received: 14 September 2000 / Published online: 18 January 2002     

Weighted Bergman spaces¶associated with causal symmetric spaces

Let H\G be a causal symmetric space sitting inside its complexification H _ℂ\G _ℂ. Then there exist certain G-invariant Stein subdomains Ξ of H _ℂ\G _ℂ. The Haar measure on H _ℂ\G _ℂ gives rise to a G-invariant measure on Ξ. With respect to this measure one can define the Bergman space B ²(Ξ) of square integrable holomorphic functions on Ξ. The group G acts unitarily on the Hilbert space B ²(Ξ) by left translations in the arguments. The main result of this paper is the Plancherel Theorem for B ²(Ξ), i.e., the disintegration formula for the left regular representation into irreducibles. Received: Received: 23 November 1998     

The Schwartz Space and Homogeneous Distributions on H-Type Groups

 Let N be an H-type group of homogeneous dimension Q. We study the space of biradial Schwartz functions on N by means of the Gelfand transform. This enables us to characterize the class of biradial homogeneous distributions on N of degree α, with 0 ? α< Q, which are away from the identity, via the Gelfand transform. (Received 26 April 2000; in revised form December 2000)     

A Two-Spheres Problem on Homogeneous Trees

m , n two relatively prime natural numbers, if a complex valued function f on a homogeneous tree satisfies the mean value property for all spheres of radius m and all spheres of radius n, then f is harmonic. Received: November 27, 1995/Revised: Revised March 16, 1998     

Measures on the Product of Compact Spaces

 If K is an uncountable metrizable compact space, we prove a “factorization” result for a wide variety of vector valued Borel measures μ defined on K ⁿ . This result essentially says that for every such measure μ there exists a measure μ′ defined on K such that the measure μ of a product A ₁ × ⋯ × A _n of Borel sets of K equals the measure μ′ of the intersection A ₁′∩⋯∩A _n ′, where the A _i ′’s are certain transforms of the A _i ’s. Partially supported by DGICYT grant PB97-0240. Received August 23, 2001; in revised form March 21, 2002     

The convolution equation of Choquet and Deny on [IN]-groups

Let be a probability measure on a locally compact groupG. A real Borel functionf onG is called -harmonic if it satisfies the convolution equation *f=f. Given that isnonsingular with its translates, we show that the bounded -harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If is nondegenerate and absolutely continuous, we solve the more general equation *= for positive measure on those groups which are metrizable and separable.Supported by Hong Kong RGC Earmarked Grant and CUHK Direct Grant     

Algebraic varieties on which the classical Phragmén-Lindelöf estimates hold for plurisubharmonic functions

Algebraic varieties V are investigated on which the natural analogue of the classical Phragmén-Lindel?f principle for plurisubharmonic functions holds. For a homogeneous polynomial P in three variables it is shown that its graph has this property if and only if P has real coefficients, no elliptic factors, is locally hyperbolic in all real characteristics, and the localizations in these characteristics are square-free. The last condition is shown to be necessary in any dimension. Received July 23, 1998     

Hörmander Multipliers on the Heisenberg Group

 Let be the Heisenberg group of dimension . Let be the partial sub-Laplacians on and T the central element of the Lie algebra of . For any we prove that the operator is bounded on the Hardy spaces , if the function m satisfies a Hrmander-type condition on which depends on . We also obtain analogous results for the operators and , where the function m satisfies analogous H?rmander-type conditions on and on , respectively. Here is the Kohn-Laplacian on . (Received 28 July 1999; in final form 6 March 2000)     

On the Zero-Divergence of Equidistant Lagrange Interpolation

 In 1942, P. Szász published the surprising result that if a function f is of bounded variation on [−1, 1] and continuous at 0 then the sequence of the equidistant Lagrange interpolation polynomials converges at 0 to . In the present note we give a construction of a function continuous on [−1, 1] whose Lagrange polynomials diverge at 0. Moreover, we show that the rate of divergence attains almost the maximal possible rate. (Received 2 February 2000)     

Weighted Polynomials on Discrete Sets

 For a real interval I of positive length, we prove a necessary and sufficient condition which ensures that the continuous L _p (0 < p ⩽ ∞) norm of a weighted polynomial, P _n w ⁿ , deg P _n  ⩽ n, n ⩾ 1 is in an nth root sense, controlled by its corresponding discrete H?lder norm on a very general class of discrete subsets of I. As a by product of our main result, we establish inequalities and theorems dealing with zero distribution, zero location and sup and L _p infinite–finite range inequalities. Received April 4, 2001; in final form June 21, 2002     

The h-p version of the coupling of finite element and boundary element methods for transmission problems in polyhedral domains

Summary. This paper analyzes the rate of convergence of the h-p version of the coupling of the finite element and boundary element method for transmission problems with a linear differential operator with variable coefficients in a bounded polyhedral domain and with constant coefficients in the exterior domain . This procedure uses the variational formulation of the differential equation in and involves integral operators on the interface between and . The finite elements are used to obtain approximate solutions of the differential equation in and the boundary elements are used to obtain approximate solutions of the integral equations. For given piecewise analytic data we show that the Galerkin solution of this coupling procedure converges exponentially fast in the energy norm if the h-p version is used both for finite elements and boundary elements. Received February 10, 1996 / Revised version received April 4, 1997     

Spectral asymptotics of periodic elliptic operators

We demonstrate that the structure of complex second-order strongly elliptic operators H on with coefficients invariant under translation by can be analyzed through decomposition in terms of versions , , of H with z-periodic boundary conditions acting on where . If the s emigroup S generated by H has a H?lder continuous integral kernel satisfying Gaussian bounds then the semigroups generated by the have kernels with similar properties and extends to a function on which is analytic with respect to the trace norm. The sequence of semigroups obtained by rescaling the coefficients of by converges in trace norm to the semigroup generated by the homogenization of . These convergence properties allow asymptotic analysis of the spectrum of H. Received September 1, 1998; in final form January 14, 1999     

Homogeneous nilmanifolds attached to representations of compact Lie groups

For each compact Lie algebra ? and each real representation V of ? we consider a two-step nilpotent Lie group N(?,V), endowed with a natural left-invariant riemannian metric. The homogeneous nilmanifolds so obtained are precisely those which are naturally reductive. We study some geometric aspects of these manifolds, finding many parallels with H-type groups. We also obtain, within the class of manifolds N(?,V), the first examples of non-weakly symmetric, naturally reductive spaces and new examples of non-commutative naturally reductive spaces. Received: 16 September 1998 / Revised version: 24 February 1999     

Polynomials on the Cauchy circle

Summary. Let be a complex polynomial of degree with and Cauchy radius 1 about the origin. We discuss the order of magnitude of the minimal number such that Previous estimates of are improved to . Some other related properties of these polynomials are also exhibited. Received March 3, 1993     

Spaces of distributions, interpolation by translates of a basis function and error estimates

Summary. Interpolation with translates of a basis function is a common process in approximation theory. The most elementary form of the interpolant consists of a linear combination of all translates by interpolation points of a single basis function. Frequently, low degree polynomials are added to the interpolant. One of the significant features of this type of interpolant is that it is often the solution of a variational problem. In this paper we concentrate on developing a wide variety of spaces for which a variational theory is available. For each of these spaces, we show that there is a natural choice of basis function. We also show how the theory leads to efficient ways of calculating the interpolant and to new error estimates. Received December 10, 1996 / Revised version received August 29, 1997     

Corrected collocation methods for periodic pseudodifferential equations

Summary. We propose an approximation method for periodic pseudodifferential equations, which yields higher convergence rates in Sobolev spaces with negative order than the collocation method. The main idea consists in correcting the usual collocation solution in a certain way by the solution of a small Galerkin system for the same equation. Both trigonometric and spline approximation methods are considered. In most of the cases our convergence result even improves that of the qualocation method. Received January 3, 1994 / Revised version received August 17, 1994     

Certain classes of pluricomplex Green functions on ${\mathbb C}^n$

We consider (pluricomplex) Green functions defined on , with logarithmic poles in a finite set and with logarithmic growth at infinity. For certain sets, we describe all the corresponding Green functions. The set of these functions is large and it carries a certain algebraic structure. We also show that for some sets no such Green functions exist. Our results indicate the fact that the set of poles should have certain algebro-geometric properties in order for these Green functions to exist. Received November 24, 1998; in final form April 19, 1999 / Published online July 3, 2000