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CATEGORIES:Applied and Computational Analysis
SUMMARY:Computing with Fourier series approximations on ge
neral domains - Daan Huybrechs (KU Leuven)
DTSTART;TZID=Europe/London:20140603T150000
DTEND;TZID=Europe/London:20140603T160000
UID:TALK51551AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/51551
DESCRIPTION:Fourier series approximations can be constructed a
nd manipulated efficiently\, and in a numerically
stable manner\, with the Fast Fourier transform (F
FT). However\, high accuracy is achieved only for
smooth and periodic functions due to the Gibbs phe
nomenon. This is a limiting factor already for fun
ctions defined on an interval\, but is even more r
estrictive for functions defined on domains with g
eneral shapes in more than one dimension. For many
domains\, it is not even clear what periodicity i
s. We show that these restrictions originate at le
ast partially in the desire to construct a basis f
or a finite-dimensional function space in which to
approximate functions. This stringent condition e
nsures uniqueness of the representation of any fun
ction in that space\, but that is not essential fo
r high-accuracy approximations. We relax the notio
n of a basis to that of a frame\, a set of functio
ns that is possibly redundant. Frames based on Fou
rier series are easily defined for very general do
mains\, and the FFT may still be used to manipulat
e the corresponding approximations. We illustrate
the surprising flexibility and approximation power
of Fourier-based frames with a variety of example
s. The corresponding algorithms are inherently ill
-conditioned due to the redundancy of the frame. Y
et\, all computations are numerically stable and a
newly developed theory proves this point.
LOCATION:MR 14\, CMS
CONTACT:
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