It is well known that a non-constant complex-valued function
f
defined on the open unit disk
D
of the complex plane is an analytic self-mapping of
D
if and only if Pick matrices
(
1
-
f
(
z
i
)
f
(
z
...j
)
¯
)
/
(
1
-
z
i
z
¯
j
)
i
,
j
=
1
n
are positive semidefinite for all choices of finitely many points
z
i
∈
D
. A stronger version of the “if” part was established by Hindmarsh (Pac J Math 27:527–531,
1968
): if all 3 × 3 Pick matrices are positive semidefinite, then
f
is an analytic self-mapping of
D
. In this paper, we extend this result to the non-commutative setting of power series over quaternions.
In this paper we establish the existence of “almost universal” quaternionic power series and entire functions. Denoting by B(0, 1) the open unit ball in H, this means that there exists a quaternionic ...power series S(q)=∑k=0∞qkak with radius of convergence 1 such that, denoting by Sn(q) the n‐th partial sum ∑k=0nqkak of S, for every K∈FH∖B(0;1)¯, for every axially symmetric open subset Ω of H containing K and every f slice regular on Ω, there exists a subsequence (Snk(q))k∈N of the partial sums of S such that Snk(q)→f(q) uniformly on K, as k→∞. The symbol FH∖B(0;1)¯ denotes the set of axially symmetric compact sets in H∖B(0;1)¯ such that CI∖(K∩CI) is connected for some I∈S. This is a slightly weaker property than the classical universal power series phenomenon obtained for f:K→C analytic only on the interior of K and continuous on K. We also generalize a result originally proven by Birkhoff and finally we show that there exists an entire quaternionic function whose set of derivatives is dense in the class of entire quaternionic functions.
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