Geodesic
Computing discrete convolutions with verified accuracy via Banach algebras and the FFT
Applications of Mathematics, Tome 63 (2018) no. 3, pp. 219-235.

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We introduce a method to compute rigorous component-wise enclosures of discrete convolutions using the fast Fourier transform, the properties of Banach algebras, and interval arithmetic. The purpose of this new approach is to improve the implementation and the applicability of computer-assisted proofs performed in weighed $\ell ^1$ Banach algebras of Fourier/Chebyshev sequences, whose norms are known to be numerically unstable. We introduce some application examples, in particular a rigorous aposteriori error analysis for a steady state in the quintic Swift-Hohenberg PDE.
DOI : 10.21136/AM.2018.0082-18
Classification : 42B05, 46B45, 46J15, 65G40, 65R10, 65T50
Keywords: discrete convolutions; Banach algebras; fast Fourier transform; interval arithmetic; rigorously verified numerics; quintic Swift-Hohenberg PDE 
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     title = {Computing discrete convolutions with verified accuracy via {Banach} algebras and the {FFT}},
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Lessard, Jean-Philippe. Computing discrete convolutions with verified accuracy via Banach algebras and the FFT. Applications of Mathematics, Tome 63 (2018) no. 3, pp. 219-235. doi : 10.21136/AM.2018.0082-18. http://geodesic.mathdoc.fr/articles/10.21136/AM.2018.0082-18/

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