# 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:
- AC02-06CH11357; RG160236; DTRA-DMS 1322393

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- SIAM Journal on Scientific Computing

- Additional Journal Information:
- Journal Volume: 39; Journal Issue: 6; Journal ID: ISSN 1064-8275

- 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. doi: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.
doi:10.1137/17M1114764.
```

```
@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 = {Thu Dec 21 00:00:00 EST 2017},

month = {Thu Dec 21 00:00:00 EST 2017}

}

Free Publicly Available Full Text

This content will become publicly available on December 21, 2018

Publisher's Version of Record

DOI: 10.1137/17M1114764

Other availability

Cited by: 1 work

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