247 research outputs found
Differential forms and Clifford analysis
In this paper we use a calculus of differential forms which is defined using an axiomatic approach. We then define integration of differential forms over chains in a new way and we present a short proof of Stokes' formula using distributional techniques. We also consider differential forms in Clifford analysis, vector differentials and their powers.
This framework enables an easy proof for a Cauchy's formula on a k-surface. Finally, we discuss how to compute winding numbers in terms of the monogenic Cauchy kernel and the vector differentials with a new approach which does not involve cohomology of differential forms
Approximation by polynomials on quaternionic compact sets
In this paper we obtain several extensions to the quaternionic setting of
some results concerning the approximation by polynomials of functions
continuous on a compact set and holomorphic in its interior. The results
include approximation on compact starlike sets and compact axially symmetric
sets. The cases of some concrete particular sets are described in details,
including quantitative estimates too
Pontryagin de Branges Rovnyak spaces of slice hyperholomorphic functions
We study reproducing kernel Hilbert and Pontryagin spaces of slice
hyperholomorphic functions which are analogs of the Hilbert spaces of analytic
functions introduced by de Branges and Rovnyak. In the first part of the paper
we focus on the case of Hilbert spaces, and introduce in particular a version
of the Hardy space. Then we define Blaschke factors and Blaschke products and
we consider an interpolation problem. In the second part of the paper we turn
to the case of Pontryagin spaces. We first prove some results from the theory
of Pontryagin spaces in the quaternionic setting and, in particular, a theorem
of Shmulyan on densely defined contractive linear relations. We then study
realizations of generalized Schur functions and of generalized Carath'eodory
functions
A Cauchy kernel for slice regular functions
In this paper we show how to construct a regular, non commutative Cauchy
kernel for slice regular quaternionic functions. We prove an (algebraic)
representation formula for such functions, which leads to a new Cauchy formula.
We find the expression of the derivatives of a regular function in terms of the
powers of the Cauchy kernel, and we present several other consequent results
Entire slice regular functions
Entire functions in one complex variable are extremely relevant in several
areas ranging from the study of convolution equations to special functions. An
analog of entire functions in the quaternionic setting can be defined in the
slice regular setting, a framework which includes polynomials and power series
of the quaternionic variable. In the first chapters of this work we introduce
and discuss the algebra and the analysis of slice regular functions. In
addition to offering a self-contained introduction to the theory of
slice-regular functions, these chapters also contain a few new results (for
example we complete the discussion on lower bounds for slice regular functions
initiated with the Ehrenpreis-Malgrange, by adding a brand new Cartan-type
theorem).
The core of the work is Chapter 5, where we study the growth of entire slice
regular functions, and we show how such growth is related to the coefficients
of the power series expansions that these functions have. It should be noted
that the proofs we offer are not simple reconstructions of the holomorphic
case. Indeed, the non-commutative setting creates a series of non-trivial
problems. Also the counting of the zeros is not trivial because of the presence
of spherical zeros which have infinite cardinality. We prove the analog of
Jensen and Carath\'eodory theorems in this setting
Fueter's theorem for monogenic functions in biaxial symmetric domains
In this paper we generalize the result on Fueter's theorem from [10] by
Eelbode et al. to the case of monogenic functions in biaxially symmetric
domains. To obtain this result, Eelbode et al. used representation theory
methods but their result also follows from a direct calculus we established in
our paper [21]. In this paper we first generalize [21] to the biaxial case and
derive the main result from that.Comment: 11 page
On two-sided monogenic functions of axial type
In this paper we study two-sided (left and right) axially symmetric solutions
of a generalized Cauchy-Riemann operator. We present three methods to obtain
special solutions: via the Cauchy-Kowalevski extension theorem, via plane wave
integrals and Funk-Hecke's formula and via primitivation. Each of these methods
is effective enough to generate all the polynomial solutions.Comment: 17 pages, accepted for publication in Moscow Mathematical Journa
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