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SUMMARY:Computing with Fourier series approximations on general domains - 
 Daan Huybrechs (KU Leuven)
DTSTART:20140603T140000Z
DTEND:20140603T150000Z
UID:TALK51551@talks.cam.ac.uk
CONTACT:6743
DESCRIPTION:Fourier series approximations can be constructed and manipulat
 ed efficiently\, and in a numerically stable manner\, with the Fast Fourie
 r transform (FFT). However\, high accuracy is achieved only for smooth and
  periodic functions due to the Gibbs phenomenon. This is a limiting factor
  already for functions defined on an interval\, but is even more restricti
 ve for functions defined on domains with general shapes in more than one d
 imension. For many domains\, it is not even clear what periodicity is. We 
 show that these restrictions originate at least partially in the desire to
  construct a basis for a finite-dimensional function space in which to app
 roximate functions. This stringent condition ensures uniqueness of the rep
 resentation of any function in that space\, but that is not essential for 
 high-accuracy approximations. We relax the notion of a basis to that of a 
 frame\, a set of functions that is possibly redundant. Frames based on Fou
 rier series are easily defined for very general domains\, and the FFT may 
 still be used to manipulate the corresponding approximations. We illustrat
 e the surprising flexibility and approximation power of Fourier-based fram
 es with a variety of examples. The corresponding algorithms are inherently
  ill-conditioned due to the redundancy of the frame. Yet\, all computation
 s are numerically stable and a newly developed theory proves this point.
LOCATION:MR 14\, CMS
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