Theory Of Regular Functions Research Articles

Дифференцирование функций кватернионной переменной

В данной работе рассматривается определение дифференцируемости и регулярностипо Фютеру [1–2] и примеры регулярных по Фютеру функций, приводится и определениеС-регулярности и С-производной или производной Куллена [3], на основе которой строитсяновая теория регулярных функций в [4], которая уже включает полиномы и сходящиеся ря-ды гиперкомплексной переменной как дифференцируемые функции. Затем предлагаетсяновое определение дифференцируемости, имеющее классический вид, но со специфиче-ской сходимостью, которое позволяет доказать теоремы о дифференцируемости суммы ипроизведения дифференцируемых функций, о дифференцируемости “частного” дифферен-цируемых функций. Далее выводится производная степени и доказывается дифференци-руемость полиномов и степенных рядов, что позволяет строить обобщения элементарныхфункций для кватернионных аргументов. Приводится пример, показывающий, что без спе-цифической сходимости приведенное определение дифференцируемости теряет смысл. Спомощью степенных рядов задаются функции, которые являются решениями дифферен-циальных уравнений с постоянными кватернионными коэффициентами. Рассматриваетсязадача отыскания корней квадратного уравнения с кватернионными коэффициентами, ко-торая возникает при решении дифференциальных уравнений

On Octonionic Regular Functions and the Szegö Projection on the Octonionic Heisenberg Group

We discuss the octonionic regular functions and the octonionic regular operator on the octonionic Heisenberg group. This is the octonionic version of CR function theory in the theory of several complex variables and regular function theory on the quaternionic Heisenberg group. By identifying the octonionic algebra with \(\mathbb{R }^{8}\), we can write the octonionic regular operator and the associated Laplacian operator as real \((8\times 8)\)-matrix differential operators. Then we use the group Fourier transform on the octonionic Heisenberg group to analyze the associated Laplacian operator and to construct its kernel. This kernel is exactly the Szego kernel of the orthonormal projection from the space of \(L^{2}\) functions to the space of \(L^{2}\) regular functions on the octonionic Heisenberg group.

Singularities of slice regular functions

AbstractBeginning in 2006, G. Gentili and D. C. Struppa developed a theory of regular quaternionic functions with properties that recall classical results in complex analysis. For instance, in each Euclidean ball B(0, R) centered at 0 the set of regular functions coincides with that of quaternionic power series \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\sum _{n \in {\mathbb {N}}} q^n a_n$\end{document} converging in B(0, R). In 2009 the author proposed a classification of singularities of regular functions as removable, essential or as poles and studied poles by constructing the ring of quotients. In that article, not only the statements, but also the proving techniques were confined to the special case of balls B(0, R). Quite recently, F. Colombo, G. Gentili and I. Sabadini (2010) and the same authors in collaboration with D. C. Struppa (2009) identified a larger class of domains, on which the theory of regular functions is natural and not limited to quaternionic power series. The present article studies singularities in this new context, beginning with the construction of the ring of quotients and of Laurent‐type expansions at points p other than the origin. These expansions, which differ significantly from their complex analogs, allow a classification of singularities that is consistent with the one given in 2009. Poles are studied, as well as essential singularities, for which a version of the Casorati‐Weierstrass Theorem is proven.

Rigidity for regular functions over Hamilton and Cayley numbers and a boundary Schwarz' Lemma

A new theory of regular functions over the skew field of Hamilton numbers (quaternions) and in the division algebra of Cayley numbers (octonions) has been recently introduced by Gentili and Struppa (Adv. Math. 216 (2007) 279–301). For these functions, among several basic results, the analogue of the classical Schwarz' Lemma has been already obtained. In this paper, following an interesting approach adopted by Burns and Krantz in the holomorphic setting, we prove some boundary versions of the Schwarz' Lemma and Cartan's Uniqueness Theorem for regular functions. We are also able to extend to the case of regular functions most of the related “rigidity” results known for holomorphic functions.

Some applications of integrated integral operators and Cauchy–Pompeiu representations in Clifford analysis†

By using the higher-order Cauchy–Pompeiu formula for functions taking values in a Clifford algebra, the class of generalized regular functions is defined. Some properties, such as the Cauchy integral representation, the Cauchy theorem, Momera's theorem, total analyticity and extension theorem of Hartog's type from theory of regular functions in Clifford analysis, are generalized for those functions. †Dedicated to Professor H. Begehr on his 65th Birthday.

Lusin area functions on local fields

1* By a local field, we mean a locally compact, nondiscrete, totally disconnected, (complete) field. Various aspects of harmonic analysis on local fields have been studied. A list of references can be found in [4]. We also refer to [4] for notation and preliminaries. Let if be a fixed local field with the ring of integers £?. &\& = GF(q) where & is the maximal ideal in & and q is a prime power. For keZ, let ^~k = {xe if: £ qk), {έ? = ^°). ^*r* = V + ^~h are spheres. The Haar measure on K has been normalized so that |<^| = dx = 1 and |^V| = qk for all k. The theory of regular functions which are the local field analogue of harmonic functions is studied in [10] and [4]. In particular, distributions on K have been identified with regular functions on if x Z and the regularization kernel Rk(x) = q~kΦ-k{x)y where Φ_fe is the characteristic function of ^~k, serves as the Poisson kernel. Write (&*y k) = {{x, k)eK x Z:xe ^~1}. For a nonnegative integer ί and ^e K, let Γι{z) = {(x, k)e K x Z: \x - z ^ qk+ι} = U/fc (^~(*+l), A;). For a distribution / on if or a regular function /(a?, k) on if x Z, denote dkf(x) — f(x, k) — f(x, k + 1). The Lusin area function of / with respect to Γt is given by

On the theory of regular functions in banach algebras

On the theory of regular functions in banach algebras