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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Optimal reconstruction of functions from their tru
ncated power series at a point - Ovidiu Costin (Oh
io State University)
DTSTART;TZID=Europe/London:20210401T160000
DTEND;TZID=Europe/London:20210401T170000
UID:TALK158455AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/158455
DESCRIPTION:
I will speak about the quest
ion of the mathematically
optimal reconstr
uction of a function from a finite number of terms
of its power
series at a point\, and on a
ditional data such as: as domain of analyticity\,<
br>
bounds or others.
 \
;
Aside from its intrinsic mathem
atical interest\, this
question is importa
nt in a variety of applications in mathematics and
physics
such as the practical computation
of the Painleve transcendents\, which I will
<
br> use as an example\, and the reconstruction of
functions from resurgent
perturbative seri
es in models of quantum field theory and string th
eory. Given
a class of functions which hav
e a common Riemann surface and a common type of bo
unds
on it\, we show that the optimal proc
edure stems from the uniformization
theore
m. A priori Riemann surface information and bounds
exist for the Borel
transform of asymptot
ic expansions in wide classes of mathematical prob
lems
such as meromorphic systems of linear
or nonlinear ODEs\, classes of PDEs and
m
any others\, \; known\, by mathematical
theorems\, \; to be resurgent. \; I will
also discuss some (apparently) new
unifor
mization methods and maps. Explicit uniformization
in Borel plane is
possible for all linear
or nonlinear second order meromorphic ODEs.
 \;
This optimal
procedure is dramatically superior to the
existing (generally ad-hoc) ones\, both theoretica
lly and in their effective
numerical appli
cation\, which I will illustrate. The comparison w
ith Pade approximants
is especially intere
sting.
 \;
W
hen more specific information exists\, such as the
nature
of the singularities of the functi
ons of interest\, we found methods based on
convolution operators to eliminate these singula
rities. The type of
singularities is known
for resurgent functions coming from many problems
in
analysis. With this addition\, the acc
uracy is improved substantially with
respe
ct to the optimal accuracy which would be possible
in full generality.
 \;
<
br>
Work in collaboration with G. Dunne\,
U. Conn.
LOCATION:Seminar Room 2\, Newton Institute
CONTACT:INI IT
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