Harmonic functions on hypergroups

We initiate a study of harmonic functions on hypergroups. In particular, we introduce the concept of a nilpotent hypergroup and show such hypergroup admits an invariant measure as well as a Liouville theorem for bounded harmonic functions. Further, positive harmonic functions on nilpotent hypergroups are shown to be integrals of exponential functions. For arbitrary hypergroups, we derive a Harnack inequality for positive harmonic functions and prove a Liouville theorem for compact hypergroups. We discuss an application to harmonic spherical functions.     

Hypergroupes de typeC

Some results on the right hypergroups of typeC are given. This class of hypergroups containD-hypergroups and cogroups (Eaton and Utumi). Connections between groups orD-hypergroups and right hypergroups of typeC are studied. Lastly all right hypergroups of typeC of cordinality smaller than five are determined (they are allD-hypergroups). Travail accompli avec l’aide du M.P.I.     

The existence and convergence of subsequences of Padé approximants

We prove the existence of an infinite number of Padé approximants, and thereby remedy a defect in Nuttall's theorem. It is proved that the sequences of Padé approximants shown by Perron, Gammel, and Wallin to be everywhere divergent contain subsequences which are everywhere convergent. It is further proved that there always exist, for entire functions, everywhere convergent [1, N] and [2, N] subsequences of Padé approximants. There must exist subsequences of [m, N] Padé approximants (N → ∞) which converge almost everywhere in $\text{¦z¦ ? ? < R}$ to functions f(z) which are regular except for a finite number (n ? m) of poles in $\text{¦z¦ < R}$. We prove convergence of the [N, N + j] Padé approximants in the mean on the Riemann sphere for meromorphic functions.     

Bessel Convolutions on Matrix Cones: Algebraic Properties and Random Walks

Bessel-type convolution algebras of measures on the matrix cones of positive semidefinite q×q-matrices over ?,?,? were introduced recently by Rösler. These convolutions depend on a continuous parameter, generate commutative hypergroups, and have Bessel functions of matrix argument as characters. In this paper, we study the algebraic structure of these hypergroups. In particular, the subhypergroups, quotients, and automorphisms are classified. The algebraic properties are partially related to the properties of random walks on these matrix Bessel hypergroups. In particular, known properties of Wishart distributions, which form Gaussian convolution semigroups on these hypergroups, are put into a new light. Moreover, limit theorems for random walks are presented. In particular, we obtain strong laws of large numbers and a central limit theorem with Wishart distributions as limits.     

Non existence of quasi-harmonic spheres

Let M and N be compact Riemannian manifolds. To prove the global existence and convergence of the heat flow for harmonic maps between M and N, it suffices to show the nonexistence of harmonic spheres and nonexistence of quasi-harmonic spheres. In this paper, we prove that, if the universal covering of N admits a nonnegative strictly convex function with polynomial growth, then there are no quasi-harmonic spheres nor harmonic spheres. This generalizes the famous Eells–Sampson’s theorem (Am J Math 86:109–169, [7]).     

Sugli ipergruppi canonici finiti con identità parziali scalari

The author analyses finite canonical hypergroups which satisfy the condition (noted i.p.s.):xoy∈x→xoy=x. He finds those of order <5 and gives examples of (i.p.s.)-hypergroups which are not strongly canonical.     

Relations and homomorphisms of n-hypergroups

In this paper, some types of homomorphisms of $n$-hypergroups is introduced and several properties are found and examples are presented. Homomorphisms of $n$-hypergroups are a generalization of homomorphisms of hypergroups. Homomorphism between $n$-hypergroups and equivalence relations on $n$-hypergroups are closely related. Also, we consider an equivalence relation $\rho$ on an $n$-hypergroup $H$ and define an $n$-hyperoperation on $H/\rho$ and prove some results in this respect.     

On Complementable Semihypergroups

In this article, we first introduce the notion of complementable semihypergroup, proving that the classes of simplifiable semigroups, groups, simplifiable semihypergroups, and complete hypergroups are examples of complementable semihypergroups. Then we define when two semihypergroups are disjoint and find examples of such semihypergroups. Finally, we discuss on the complementable property of K_H-hypergroups.     

On the dual of a commutative signed hypergroup

Signed hypergroups are convolution structures similar to hypergroups, though being not necessarily positivity-preserving. We prove a generalized Plancherel theorem for positive definite measures on a commutative signed hypergroup, with an analogue of the classical Plancherel theorem as a special case. Moreover, signed hypergroups with subexponential growth are studied. As an application, the dual of the Laguerre convolution structure on ℝ₊ is determined.     

Recent Developments In Harmonic Approximation,With Applications

A theorem of J.L. Walsh (1929) says that if E is a compact subset of Rⁿ with connected complement and if u is harmonic on a neighbourhood of E, then u can be uniformly approximated on E by functions harmonic on the whole of Rⁿ. In Part I of this article we survey some generalizations of Walsh’s theorem from the period 1980–94. In Part II we discuss applications of Walsh’s theorem and its generalizations to four diverse topics: universal harmonic functions, the Radon transform, the maximum principle, and the Dirichlet problem.     

Frontière de Martin des marches aléatoires sur certains hypergroupes d-dimensionnelsMartin boundary of random walks on certain d-dimensional hypergroups

We give an integral representation formula for harmonic functions of Markov chains on $\text{N}{}^{\mathrm{d}}$ and $\text{R}{}_{\mathrm{+}}{}^{\mathrm{d}}$ which transition probability is invariant by translations of a hypergroup, product of polynomial hypergroups for $\text{N}{}^{\mathrm{d}}$ and product of Sturm–Liouville hypergroups for $\text{R}{}_{\mathrm{+}}{}^{\mathrm{d}}$. To cite this article: L. Godefroy, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1029–1034.     

Boolean functions with MacWilliams duality

We introduce a new class of Boolean functions for which the MacWilliams duality holds, called MacWilliams-dual functions, by considering a dual notion on Boolean functions. By using the MacWilliams duality, we prove the Gleason-type theorem on MacWilliams-dual functions. We show that a collection of MacWilliams-dual functions contains all the bent functions and all formally self-dual functions. We also obtain the Pless power moments for MacWilliams-dual functions. Furthermore, as an application, we prove the nonexistence of bent functions in 2n variables with minimum degree n?k for any nonnegative integer k and n ≥ N with some positive integer N under a certain condition.     

The Oshima-Sekiguchi and Liouville theorems on Heintze groups

Let L be an elliptic operator on a Riemannian manifold M. A function F annihilated by L is said to be L-harmonic. F is said to have moderate growth if and only if F grows at most exponentially in the Riemannian distance. If M is a rank-one symmetric space and L is the Laplace-Beltrami operator for M, the Oshima-Sekiguchi theorem [T. Oshima, J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980) 1-81] states that a L-harmonic function F has moderate growth if and only if F is the Poisson integral of a distribution on the Furstenberg boundary. In this work we prove that this result generalizes to a very large class of homogeneous Riemannian manifolds of negative curvature. We also (i) prove a Liouville type theorem that characterizes the “polynomial-like” harmonic functions which vanish on the boundary in terms of their growth properties, (ii) describe all “polynomial-like” harmonic functions, and (iii) give asymptotic expansions for the Poisson kernel. One consequence of this work is that every Schwartz distribution on the boundary is the boundary value for a L-harmonic function F which is uniquely determined modulo “polynomial-like” harmonic functions.     

Moment Functions and Central Limit Theorem for Jacobi Hypergroups on [0,\infty [

In this paper, we derive sharp estimates and asymptotic results for moment functions on Jacobi type hypergroups. Moreover, we use these estimates to prove a central limit theorem (CLT) for random walks on Jacobi hypergroups with growing parameters $\alpha ,\beta \rightarrow \infty $ . As a special case, we obtain a CLT for random walks on the hyperbolic spaces ${H}_d(\mathbb F )$ with growing dimensions $d$ over the fields $\mathbb F =\mathbb R ,\ \mathbb C $ or the quaternions $\mathbb H $ .     

L q harmonic functions on graphs

We prove an analogue of Yau’s Caccioppoli-type inequality for nonnegative subharmonic functions on graphs. We then obtain a Liouville theorem for harmonic or nonnegative subharmonic functions of class L ^q , 1 ≤ q < ∞, on any graph, and a quantitative version for q > 1. Also, we provide counterexamples for Liouville theorems for 0 < q < 1.     

A local mean value theorem for functions on non-archimedean field extensions of the real numbers

In this paper, we review the definition and properties of locally uniformly differentiable functions on N, a non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. Then we define and study n-times locally uniform differentiable functions at a point or on a subset of N. In particular, we study the properties of twice locally uniformly differentiable functions and we formulate and prove a local mean value theorem for such functions.     

Some Riemann boundary value problems in Clifford analysis

In this paper, we mainly study the R_m (m>0) Riemann boundary value problems for functions with values in a Clifford algebra C?(V_{3, 3}). We prove a generalized Liouville‐type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k‐monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the R_m (m>0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright © 2009 John Wiley & Sons, Ltd.     

On Harmonic Diffeomorphisms of the Unit Disc onto a Convex Domain

We prove a theorem for harmonic diffeomorphisms between the unit disc and a convex Jordan domain, which is a generalization of Heinz theorem [E. Heinz (1959). On one-to-one harmonic mappings. Pacific J . Math ., 9 , 101-105] for harmonic diffeomorphisms of the unit disc onto itself. We give a number of corollaries of the theorem we prove.