slice hyperholomorphic functions
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2021 ◽  
Vol 31 (3) ◽  
Author(s):  
Fabrizio Colombo ◽  
Jonathan Gantner ◽  
Stefano Pinton

AbstractThe aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for n-tuples of operators $$(A_1,\ldots ,A_n)$$ ( A 1 , … , A n ) . A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the S-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the F-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S-spectrum.



2020 ◽  
pp. 1-24
Author(s):  
Daniel Alpay ◽  
Kamal Diki ◽  
Irene Sabadini

In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions. We investigate also an extension version of the Fueter mapping theorem in this polyanalytic setting. In particular, we show that under axially symmetric conditions it is always possible to construct Fueter regular and poly-Fueter regular functions through slice polyanalytic ones using what we call the poly-Fueter mappings. We study also some integral representations of these results on the quaternionic unit ball.



2020 ◽  
Vol 31 (4) ◽  
pp. 607-628
Author(s):  
Daniel Alpay ◽  
Fabrizio Colombo ◽  
Irene Sabadini




Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1197-1207
Author(s):  
Sanjay Kumar ◽  
S.D. Sharma ◽  
Khalid Manzoor

In this paper, we define the quaternionic Fock spaces F p? of entire slice hyperholomorphic functions in a quaternionic unit ball B in H: We also study growth estimates and various results of entire slice regular functions in these spaces. The work of this paper is motivated by the recent work of [5] and [26].



Author(s):  
Daniel Alpay ◽  
Fabrizio Colombo ◽  
Izchak Lewkowicz ◽  
Irene Sabadini




2017 ◽  
Vol 290 (14-15) ◽  
pp. 2259-2279
Author(s):  
Jonathan Gantner ◽  
J. Oscar González-Cervantes ◽  
Tim Janssens


2016 ◽  
Vol 27 (2) ◽  
pp. 1459-1477 ◽  
Author(s):  
Sanjay Kumar ◽  
Khalid Manzoor ◽  
Pavinder Singh


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