75 research outputs found
Meromorphic extensions from small families of circles and holomorphic extensions from spheres
Let B be the open unit ball in C^2 and let a, b, c be three points in C^2
which do not lie in a complex line, such that the complex line through a and b
meets B and such that is different from 1 if one of the points a, b is in
B and the other in the complement of B and such that at least one of the
numbers , is different from 1. We prove that if a continuous
function f on the sphere bB extends holomorphically into B along each complex
line which passes through one of the points a, b, c then f extends
holomorphically through B. This generalizes recent work of L.Baracco who proved
such a result in the case when the points a, b, c are contained in B. The proof
is different from the one of Baracco and uses the following one variable result
which we also prove in the paper and which in the real analytic case follows
from the work of M.Agranovsky: Let D be the open unit disc in C. Given a in D
let C(a) be the family of all circles in D obtained as the images of circles
centered at the origin under an automorphism of D that maps the origin to a.
Given distinct points a, b in D and a positive integer n, a continuous function
f on the closed unit disc extends meromorphically from every circle T in either
C(a) or C(b) through the disc bounded by T with the only pole at the center of
T of degree not exceeding n if and only if f is of the form f(z) =
g_0(z)+g_1(z)\bar z +...+ g_n(z)\bar z^n where the functions g_0, g_1, ..., g_n
are holomorphic on D.Comment: This replaces the original version where an assumption was missing in
Corollary 1.3. This assumption is now added and in the last section an
example is added which shows that the added assumption is really necessar
The winding number of PF+1 for polynomials P and meromorphic extendibility of F
Let D be the open unit disc in C. The paper deals with the following
conjecture: If f is a continuous function on bD such that the change of
argument of Pf+1 around bD is nonnegative for every polynomial P such that Pf+1
has no zero on bD then f extends holomorphically through D. We prove a related
result on meromorphic extendibility for smooth functions with finitely many
zeros of finite order, which, in particular, implies that the conjecture holds
for real analytic functions.Comment: 13 page
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