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SUMMARY:Approximation Theory for a Rational Orthogonal Basis on the Real L
 ine - Marcus Webb (University of Manchester)
DTSTART:20191210T150000Z
DTEND:20191210T153000Z
UID:TALK135526@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:The Malmquist-Takenaka basis is a rational orthogonal basis co
 nstructed by mapping the Laurent basis from the unit circle to the real li
 ne by a M&ouml\;bius transformation and multiplying by a weight to ensure 
 orthogonality. Over the last century its properties have piqued the intere
 st of various researchers including Boyd\, Weideman\, Christov\, and Wiene
 r. Despite this history\, the approximation theory of this basis still def
 ies straightforward description. For example\, it was shown by Boyd and We
 ideman that for entire functions the convergence of approximation is super
 algebraic\, but that exponential convergence is only possible if the funct
 ion is analytic at infinity (i.e. at the top of the Riemann sphere --- qui
 te a strong condition). Nonetheless\, convergence can be surprisingly quic
 k\, and the main body of this talk will be the result that wave packets cl
 early cannot have exponentially convergent approximations\, but they /init
 ially/ exhibit exponential convergence for large wave packet frequencies w
 ith exponential convergence rate proportional to said frequency. Hence\, O
 (log(|eps|) omega) coefficients are required to resolve a wave packet to a
 n error of O(eps). The proof is by the method of steepest descent in the c
 omplex plane. This is joint work with Arieh Iserles and Karen Luong (Cambr
 idge).
LOCATION:Seminar Room 1\, Newton Institute
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