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Brownian motion and generalized analytic and inner functions

Alain Bernard, Eddy A. Campbell, A. M. Davie (1979)

Annales de l'institut Fourier

Let f be a mapping from an open set in R p into R q , with p > q . To say that f preserves Brownian motion, up to a random change of clock, means that f is harmonic and that its tangent linear mapping in proportional to a co-isometry. In the case p = 2 , q = 2 , such conditions signify that f corresponds to an analytic function of one complex variable. We study, essentially that case p = 3 , q = 2 , in which we prove in particular that such a mapping cannot be “inner” if it is not trivial. A similar result for p = 4 , q = 2 would solve...

Canonical bases for 𝔰𝔩 ( 2 , ℂ ) -modules of spherical monogenics in dimension 3

Roman Lávička (2010)

Archivum Mathematicum

Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as 𝔰𝔩 ( 2 , ℂ ) -modules. As finite-dimensional irreducible 𝔰𝔩 ( 2 , ℂ ) -modules, they have canonical bases which are, by construction, orthogonal. In this note, we show that these orthogonal bases form the Appell system and coincide with those constructed recently by S. Bock and K. Gürlebeck in [3]. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.

Carleson measure and monogenic functions

S. Bernstein, P. Cerejeiras (2007)

Studia Mathematica

We present necessary and sufficient conditions for a measure to be a p-Carleson measure, based on the Poisson and Poisson-Szegő kernels of the n-dimensional unit ball.

Casoratien et équations aux différences p -adiques

Jean-Paul Bézivin (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

Dans cet article, nous démontrons une inégalité liant la croissance d’un casoratien généralisé de m séries entières p -adique à la croissance du casoratien ordinaire de ces m séries entières. Il en résulte que si le casoratien de m fonctions entières p -adiques est un polynôme non nul, alors toutes ces fonctions sont des polynômes. Comme application, nous montrons que si une équation aux différences linéaire d’ordre t à coefficients dans ℂ p [ x ] a t solutions méromorphes dans tout ℂ p , linéairement indépendantes...

Clifford and harmonic analysis on cylinders and torii.

Rolf Sören Krausshar, John Ryan (2005)

Revista Matemática Iberoamericana

Cotangent type functions in Rn are used to construct Cauchy kernels and Green kernels on the conformally flat manifolds Rn/Zk where 1 < = k ≤ M. Basic properties of these kernels are discussed including introducing a Cauchy formula, Green's formula, Cauchy transform, Poisson kernel, Szegö kernel and Bergman kernel for certain types of domains. Singular Cauchy integrals are also introduced as are associated Plemelj projection operators. These in turn are used to study Hardy spaces in this...

Clifford-Hermite-monogenic operators

Freddy Brackx, Nele de Schepper, Frank Sommen (2006)

Czechoslovak Mathematical Journal

In this paper we consider operators acting on a subspace ℳ of the space L 2 ( ℝ m ; ℂ m ) of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace ℳ is defined as the orthogonal sum of spaces ℳ s , k of specific Clifford basis functions of L 2 ( ℝ m ; ℂ m ) . Every Clifford endomorphism of ℳ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic operators are characterized in terms of commutation relations and they...

Cluster sets of analytic multivalued functions

S. Harbottle (1992)

Studia Mathematica

Classical theorems about the cluster sets of holomorphic functions on the unit disc are extended to the more general setting of analytic multivalued functions, and examples are given to show that these extensions cannot be improved.

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