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Computing Discrete Convolutions with Verified Accuracy Via Banach Algebras and the FFT

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Abstract

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 ℓ1 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.

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Authors and Affiliations

  1. McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada

    Jean-Philippe Lessard

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  1. Jean-Philippe Lessard
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Correspondence to Jean-Philippe Lessard.

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Dedicated to the 50th birthday of Sergey Korotov

The research has been supported by NSERC.

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Lessard, JP. Computing Discrete Convolutions with Verified Accuracy Via Banach Algebras and the FFT. Appl Math 63, 219–235 (2018). https://doi.org/10.21136/AM.2018.0082-18

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  • DOI: https://doi.org/10.21136/AM.2018.0082-18

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