By Cinzia Bisi, Caterina Stoppato (auth.), Graziano Gentili, Irene Sabadini, Michael Shapiro, Franciscus Sommen, Daniele C. Struppa (eds.)

ISBN-10: 8847024447

ISBN-13: 9788847024441

ISBN-10: 8847024455

ISBN-13: 9788847024458

This quantity is meant to assemble vital learn effects to the lectures and discussions which happened in Rome, on the INdAM Workshop on diverse Notions of Regularity for capabilities of Quaternionic Variables in September 2010. This quantity will gather contemporary and new effects, that are hooked up to the subject lined in the course of the workshop. The paintings goals at bringing jointly overseas prime experts within the box of Quaternionic and Clifford research, in addition to younger researchers attracted to the topic, with the belief of featuring and discussing contemporary effects, examining new traits and methods within the zone and, in most cases, of marketing clinical collaboration. specific realization is paid to the presentation of other notions of regularity for features of hypercomplex variables, and to the research of the most positive aspects of the theories that they originate.

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**Sample text**

KΩi (·,·) is a reproducing kernel on Ωi . Let {φn }n∈N be a orthonormal total family of functions in A(Ωi ), and let Ki be a compact subset of Ωi . Then φ(z)φ(w) n≥0 sums uniformly on Ki × Ki to the Bergman kernel KΩi (z, w). The proofs of the various facts are immediate consequences of Proposition 5, but a comment is in order. The function KΩi (z, w) is C(i)-valued. The conjugation considered in (i) is the quaternionic conjugation which, in particular, restricts to the complex conjugation on each complex plane, so (i) follows.

Furthermore, on open balls centred at the origin, to have a power series expansion is a necessary and sufficient condition for a function to be slice regular. Since we will work with functions defined on open balls centred at the origin, we will make use of this characterization. Indeed, the purpose of this paper is to prove an analog of the Bloch-Landau Theorem for slice regular functions. In the complex case, this result is an important fact in the study of the range of holomorphic functions defined on the open unit disc D.

Then there exists a constant λK > 0 such that sup f (q) q ∈ K ≤ λK f A (Ω) , ∀f ∈ A (Ω). 1 given in [19], one can find a constant αK > 0 depending of K, such that sup g(z) z ∈ K ≤ αK |g|2 dσi 1 2 , Ωi for any function g belonging to the usual holomorphic Bergman space for Ωi . Since every f ∈ A (Ω) is of the form f|Ωi = f1 + f2 j, where the unit vector j is orthogonal to i, then sup f (q) q ∈ K = sup f|Ωi (q) q ∈ K + sup f2 (q) ≤ sup f1 (q) ≤ αK q ∈K |f1 | dσi 2 1 2 + αK Ωi ≤ 2αK |f2 | dσi 2 1 2 Ωi |f|Ωi | dσi 2 Ωi q ∈K 1 2 .

### Advances in Hypercomplex Analysis by Cinzia Bisi, Caterina Stoppato (auth.), Graziano Gentili, Irene Sabadini, Michael Shapiro, Franciscus Sommen, Daniele C. Struppa (eds.)

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