Strictly positive definite functions on a compact group

We recognize a result of Schreiner, concerning strictly positive definite functions on a sphere in an Euclidean space, as a generalization of Bochner's theorem for compact groups.

     

Gelfand transforms of polyradial Schwartz functions on the Heisenberg group

We prove that the Gelfand transform is a topological isomorphism between the space of polyradial Schwartz functions on the Heisenberg group and the space of Schwartz functions on the Heisenberg brush. We obtain analogous results for radial Schwartz functions on Heisenberg type groups.     

Positive definite functions on the Heisenberg group

Work supported in part by the NSF     

Lipschitz continuity of refinable functions on the Heisenberg group

In this paper, the Lipschitz continuity of refinable functions related to the general acceptable dilations on the Heisenberg group will be investigated in terms of the uniform joint spectral radius. We also give an investigation of the refinable functions in the generalized Lipschitz spaces related to a kind of special acceptable dilations.     

On differentiability and analyticity of positive definite functions

We derive a set of differential inequalities for positive definite functions based on previous results derived for positive definite kernels by purely algebraic methods. Our main results show that the global behavior of a smooth positive definite function is, to a large extent, determined solely by the sequence of even-order derivatives at the origin: if a single one of these vanishes then the function is constant; if they are all non-zero and satisfy a natural growth condition, the function is real-analytic and consequently extends holomorphically to a maximal horizontal strip of the complex plane.     

On a new condition for strictly positive definite functions on spheres

Recently, Xu and Cheney (1992) have proved that if all the Legendre coefficients of a zonal function defined on a sphere are positive then the function is strictly positive definite. It will be shown in this paper that, even if finitely many of the Legendre coefficients are zero, the strict positive definiteness can be assured. The results are based on approximation properties of singular integrals, and provide also a completely different proof of the results of Xu and Cheney.

     

Extensions of positive definite functions on free groups

An analogue of Krein's extension theorem is proved for operator-valued positive definite functions on free groups. The proof gives also the parametrization of all extensions by means of a generalized type of Szegö parameters. One singles out a distinguished completion, called central, which is related to quasi-multiplicative positive definite functions. An application is given to factorization of noncommutative polynomials.     

The growth of H-harmonic functions on the Heisenberg group

We discuss the relationship between the frequency and the growth of H-harmonic functions on the Heisenberg group.Precisely,we prove that an H-harmonic function must be a polynomial if its frequency is globally bounded.Moreover,we show that a class of H-harmonic functions are homogeneous polynomials provided that the frequency of such a function is equal to some constant.     

Gelfand pairs on the Heisenberg group and Schwartz functions

Let Hn be the (2n+1)-dimensional Heisenberg group and K a compact group of automorphisms of Hn such that (K?Hn,K) is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of K-invariant Schwartz functions on Hn and the space of Schwartz function on a closed subset of Rs homeomorphic to the Gelfand spectrum of the Banach algebra of K-invariant integrable functions on Hn.     

Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group

Gabor frames with Hermite functions are equivalent to sampling sequences in true Fock spaces of polyanalytic functions. In the L ²-case, such an equivalence follows from the unitarity of the polyanalytic Bargmann transform. We will introduce Banach spaces of polyanalytic functions and investigate the mapping properties of the polyanalytic Bargmann transform on modulation spaces. By applying the theory of coorbit spaces and localized frames to the Fock representation of the Heisenberg group, we derive explicit polyanalytic sampling theorems which can be seen as a polyanalytic version of the lattice sampling theorem discussed by J.M. Whittaker in Chapter 5 of his book Interpolatory Function Theory.     

四元Heisenberg群上次拉普拉斯算子的m幂次的基本解

本文研究了四元Heisenberg群上次拉普拉斯算子的m幂次的基本解,该结论是Heisenberg群上结果的推广.本文利用了四元Heisenberg群上的Fourier变换理论构造了该群上次拉普拉斯算子的m幂次的基本解,并且给出了基本解的积分表示.     

Solvability on the Heisenberg group

We solve in various spaces the linear equations L_αg = f , where L_α belongs to a class of transversally elliptic second order differential operators on the Heisenberg group with double characteristics and complex‐valued coefficients, not necessarily locally solvable. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

四元Heisenberg群上次拉普拉斯算子的m幂次的基本解

本文研究了四元Heisenberg群上次拉普拉斯算子的m幂次的基本解,该结论是Heisenberg群上结果的推广.本文利用了四元Heisenberg群上的Fourier变换理论构造了该群上次拉普拉斯算子的m幂次的基本解,并且给出了基本解的积分表示.     

各向异性Heisenberg群上带余项的Hardy型不等式

利用一些非常精细的估计技巧,证明了各向异性Heisenberg群上的一类带余项的Hardy型不等式,推广了最近文献中关于Heisenberg群上的带余项的Hardy型不等式的结果.     

Positive definite matrices and differentiable reproducing kernel inequalities

Let I⊆R be a interval and be a reproducing kernel on I. By the Moore-Aronszajn theorem, every finite matrix k(xi,xj) is positive semidefinite. We show that, as a direct algebraic consequence, if k(x,y) is appropriately differentiable it satisfies a 2-parameter family of differential inequalities of which the classical diagonal dominance is the order 0 case. An application of these inequalities to kernels of positive integral operators yields optimal Sobolev norm bounds.     

Hausdorff operators on the Heisenberg group

This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group H n. The sharp bounds for the strong type(p, p)(1 ≤ p ≤∞) estimates of n-dimensional Hausdorff operators on H n are obtained. The sharp bounds for strong(p, p) estimates are further extended to multilinear cases. As an application, we derive the sharp constant for the multilinear Hardy operator on H n. The weak type(p, p)(1 ≤ p ≤∞) estimates are also obtained.     

Accessible domains in the Heisenberg group

We study the problem of accessibility of boundary points for domains in the sub-Riemannian setting of the first Heisenberg group. A sufficient condition for accessibility is given. It is a Dini-type continuity condition for the horizontal gradient of the defining function. The sharpness of this condition is shown by examples.

     

Green functions and related boundary value problems on the Heisenberg group

In this article, a modified Kelvin transform on ? _n using inversion with respect to a ball of arbitrary radius is defined, which gives explicit expressions for Green's function and Poisson's kernel for the Korányi ball of arbitrary radius and annular domain. The solution of the Dirichlet problem for the union of two balls is discussed using the Schwarz's alternating method.