31 research outputs found
Singularities of slice regular functions
Beginning 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 centered at 0 the set of regular
functions coincides with that of quaternionic power series converging in the
same ball. 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
centered at 0. In a subsequent paper, F. Colombo, G. Gentili, I. Sabadini and
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 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.Comment: 25 pages, 1 figur
A local representation formula for quaternionic slice regular functions
After their introduction in 2006, quaternionic slice regular functions have
mostly been studied over domains that are symmetric with respect to the real
axis. This choice was motivated by some foundational results published in 2009,
such as the Representation Formula for axially symmetric domains.
The present work studies slice regular functions over domains that are not
axially symmetric, partly correcting the hypotheses of some previously
published results. In particular, this work includes a Local Representation
Formula valid without the symmetry hypothesis. Moreover, it determines a class
of domains, called simple, having the following property: every slice regular
function on a simple domain can be uniquely extended to the symmetric
completion of its domain.Comment: 10 pages, to appear in Proc. Amer. Math. So
A new series expansion for slice regular functions
A promising theory of quaternion-valued functions of one quaternionic
variable, now called slice regular functions, has been introduced in 2006. The
basic examples of slice regular functions are power series centered at 0 on
their balls of convergence. Conversely, if f is a slice regular function then
it admits at each point of its domain an expansion into power series, where the
powers are taken with respect to an appropriately defined multiplication *.
However, the information provided by such an expansion is somewhat limited by a
fact: if the center p of the series does not lie on the real axis then the set
of convergence needs not be a Euclidean neighborhood of p. We are now able to
construct a new type of expansion that is not affected by this phenomenon: an
expansion into series of polynomials valid in open subsets of the domain. Along
with this construction, we present applications to the computation of the
multiplicities of zeros and of partial derivatives.Comment: 20 pages, 1 figur
A unified notion of regularity in one hypercomplex variable
We define a very general notion of regularity for functions taking values in
an alternative real -algebra. Over Clifford numbers, this notion subsumes
the well-established notions of monogenic function and slice-monogenic
function. Over quaternions, in addition to subsuming the notions of
Fueter-regular function and of slice-regular function, it gives rise to an
entirely new theory, which we develop in some detail.Comment: 16 page
Completeness on the worm domain and the M\"untz-Sz\'asz problem for the Bergman space
In this paper we are concerned with the problem of completeness in the
Bergman space of the worm domain and its truncated version
. We determine some orthogonal systems and show that they are
not complete, while showing that the union of two particular of such systems is
complete.
In order to prove our completeness result we introduce the Muentz-Szasz
problem for the 1-dimensional Bergman space of the disk and find a sufficient condition for its solution.Comment: 14 pages, Author Accepted Manuscrip