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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Approximation Theory for a Rational Orthogonal Bas
is on the Real Line - Marcus Webb (University of M
anchester)
DTSTART;TZID=Europe/London:20191210T150000
DTEND;TZID=Europe/London:20191210T153000
UID:TALK135526AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/135526
DESCRIPTION:The Malmquist-Takenaka basis is a rational orthogo
nal basis constructed by mapping the Laurent basis
from the unit circle to the real line by a Mö
\;bius transformation and multiplying by a weight
to ensure orthogonality. Over the last century its
properties have piqued the interest of various re
searchers including Boyd\, Weideman\, Christov\, a
nd Wiener. Despite this history\, the approximatio
n theory of this basis still defies straightforwar
d description. For example\, it was shown by Boyd
and Weideman that for entire functions the converg
ence of approximation is superalgebraic\, but that
exponential convergence is only possible if the f
unction is analytic at infinity (i.e. at the top o
f the Riemann sphere --- quite a strong condition)
. Nonetheless\, convergence can be surprisingly qu
ick\, and the main body of this talk will be the r
esult that wave packets clearly cannot have expone
ntially convergent approximations\, but they /init
ially/ exhibit exponential convergence for large w
ave packet frequencies with exponential convergenc
e rate proportional to said frequency. Hence\, O(l
og(|eps|) omega) coefficients are required to reso
lve a wave packet to an error of O(eps). The proof
is by the method of steepest descent in the compl
ex plane. This is joint work with Arieh Iserles an
d Karen Luong (Cambridge).
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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