Asymptotic Fourier Coefficients for a C ∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem
dc.contributor.author | Boyd, John P. | en_US |
dc.date.accessioned | 2006-09-08T21:25:12Z | |
dc.date.available | 2006-09-08T21:25:12Z | |
dc.date.issued | 2005-11-17 | en_US |
dc.identifier.citation | Boyd, John P.; (2005). "Asymptotic Fourier Coefficients for a C ∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem." Journal of Scientific Computing (): 1-24. <http://hdl.handle.net/2027.42/43417> | en_US |
dc.identifier.issn | 0885-7474 | en_US |
dc.identifier.issn | 1573-7691 | en_US |
dc.identifier.uri | https://hdl.handle.net/2027.42/43417 | |
dc.description.abstract | In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval , to a function which is periodic on the larger interval . We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval , identically zero for , and varies smoothly in between. Such smoothed “top-hat” functions are “bells” in wavelet theory. Our bell is (for x ≥ 0) where where . By applying steepest descents to approximate the coefficient integrals in the limit of large degree j , we show that when the width L is fixed, the Fourier cosine coefficients a j of on are proportional to where Λ( j ) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the N th term, the width should be chosen to increase with N as . We derive similar asymptotics for the function f ( x )= x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergence | en_US |
dc.format.extent | 233801 bytes | |
dc.format.extent | 3115 bytes | |
dc.format.mimetype | application/pdf | |
dc.format.mimetype | text/plain | |
dc.language.iso | en_US | |
dc.publisher | Kluwer Academic Publishers-Plenum Publishers; Springer Science+Business Media, Inc. | en_US |
dc.subject.other | Fourier Series | en_US |
dc.subject.other | A Symptotic Fourier Coefficients | en_US |
dc.subject.other | Spectral Methods | en_US |
dc.subject.other | Local Fourier Basis | en_US |
dc.subject.other | Fourier Extension | en_US |
dc.title | Asymptotic Fourier Coefficients for a C ∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem | en_US |
dc.type | Article | en_US |
dc.subject.hlbsecondlevel | Education | en_US |
dc.subject.hlbsecondlevel | Science (General) | en_US |
dc.subject.hlbtoplevel | Science | en_US |
dc.subject.hlbtoplevel | Social Sciences | en_US |
dc.description.peerreviewed | Peer Reviewed | en_US |
dc.contributor.affiliationum | Department of Atmospheric, Oceanic and Space Science and Laboratory for Scientific Computation, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI, 48109, USA, | en_US |
dc.contributor.affiliationumcampus | Ann Arbor | en_US |
dc.description.bitstreamurl | http://deepblue.lib.umich.edu/bitstream/2027.42/43417/1/10915_2005_Article_9010.pdf | en_US |
dc.identifier.doi | http://dx.doi.org/10.1007/s10915-005-9010-7 | en_US |
dc.identifier.source | Journal of Scientific Computing | en_US |
dc.owningcollname | Interdisciplinary and Peer-Reviewed |
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