A Fast Algorithm for the Convolution of Functions with Compact Support Using Fourier Extensions
Abstract
In this paper, we present a new algorithm for computing the convolution of two compactly supported functions. The algorithm approximates the functions to be convolved using Fourier extensions and then uses the fast Fourier transform to efficiently compute Fourier extension approximations to the pieces of the result. Finally, the complexity of the algorithm is O(N(log N)^{2}), where N is the number of degrees of freedom used in each of the Fourier extensions.
 Authors:

 Univ. of Kent, Canterbury (United Kingdom). School of Mathematics, Statistics, and Actuarial Science
 Argonne National Lab. (ANL), Argonne, IL (United States). Mathematics and Computer Science Division
 Univ. of California, Davis, CA (United States). Dept. of Mathematics
 Publication Date:
 Research Org.:
 Argonne National Lab. (ANL), Argonne, IL (United States); Univ. of California, Davis, CA (United States); Univ. of Kent, Canterbury (United Kingdom)
 Sponsoring Org.:
 USDOE Office of Science (SC); National Science Foundation (NSF); Royal Society (United Kingdom)
 OSTI Identifier:
 1427516
 Grant/Contract Number:
 AC0206CH11357; RG160236; DTRADMS 1322393
 Resource Type:
 Accepted Manuscript
 Journal Name:
 SIAM Journal on Scientific Computing
 Additional Journal Information:
 Journal Volume: 39; Journal Issue: 6; Journal ID: ISSN 10648275
 Publisher:
 SIAM
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; convolution; fast Fourier transform; Fourier extension; Fredholm convolution integral; Toeplitz matrix; Volterra convolution integral
Citation Formats
Xu, Kuan, Austin, Anthony P., and Wei, Ke. A Fast Algorithm for the Convolution of Functions with Compact Support Using Fourier Extensions. United States: N. p., 2017.
Web. doi:10.1137/17M1114764.
Xu, Kuan, Austin, Anthony P., & Wei, Ke. A Fast Algorithm for the Convolution of Functions with Compact Support Using Fourier Extensions. United States. https://doi.org/10.1137/17M1114764
Xu, Kuan, Austin, Anthony P., and Wei, Ke. Thu .
"A Fast Algorithm for the Convolution of Functions with Compact Support Using Fourier Extensions". United States. https://doi.org/10.1137/17M1114764. https://www.osti.gov/servlets/purl/1427516.
@article{osti_1427516,
title = {A Fast Algorithm for the Convolution of Functions with Compact Support Using Fourier Extensions},
author = {Xu, Kuan and Austin, Anthony P. and Wei, Ke},
abstractNote = {In this paper, we present a new algorithm for computing the convolution of two compactly supported functions. The algorithm approximates the functions to be convolved using Fourier extensions and then uses the fast Fourier transform to efficiently compute Fourier extension approximations to the pieces of the result. Finally, the complexity of the algorithm is O(N(log N)2), where N is the number of degrees of freedom used in each of the Fourier extensions.},
doi = {10.1137/17M1114764},
journal = {SIAM Journal on Scientific Computing},
number = 6,
volume = 39,
place = {United States},
year = {2017},
month = {12}
}
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Figures / Tables:
Fig. 1.1: Schematic illustrating how the limits of integration in (1.1) vary with x, assuming that d − c ≥ b − a. Given a value of x on the horizontal axis, the limits range over the values of t indicated by the corresponding “slice” of the parallelogram.
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