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The Convolution as a Mathematical Object

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Research in History and Philosophy of Mathematics (CSHPM 2016)

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Abstract

For many scientists, the convolution is well known as a tool of applied mathematics dating from the 20th century. However, the idea of a discrete convolution can be traced as far back as the Middle Ages in China. After d’Alembert and Euler, applications of convolution integrals appeared at the end of the 18th century and the beginning of the 19th century. These were in potential theory, the heat conduction equation, and the wave equation as developed by Lagrange, Laplace, Legendre, Fourier, Cauchy, and Poisson. Here we emphasize Fourier’s contribution to the superposition principle for memory, time delay, and superposition of events. We consider the modern mathematical properties of the convolution with Volterra, Lebesgue, Doetsch, and Schwartz, and we illustrate some applications of the convolution in the theory of probability, interpolation, and smoothing.

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Notes

  1. 1.

    Note that several convolution integrals previously appeared in the 1805 Lacroix “Traité des différences et des séries” as the solutions of partial differential equations. For example, we found on page 506 the following convolution integrals: ξ(u, v) =  ∫ T(u − t)φ(t)dt +  ∫ T i (v − t)ψ(t)dt and a double integral on page 518:

    \( \iint \frac{{\left(u-x\right)}^n{\left(v-y\right)}^m}{\mathrm{1.2.3}\dots n.\mathrm{1.2.3}\dots m}\psi \left(x,y\right) dxdy. \) Lacroix’s book is available from the website of the Bibliothèque Nationale de France.

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Acknowledgement

The author would like to thank Mark Lewis for his help in editing this article.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Royal Military College of Canada, 92 Florence Street, Kingston, ON, Canada, K7M 1Y6

    Roger Godard

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  1. Roger Godard
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Correspondence to Roger Godard .

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Editors and Affiliations

  1. Mathematical, Information and Computer Sciences, Point Loma Nazarene University, San Diego, California, USA

    Maria Zack

  2. Department of Philosophy, McGill University, Montreal, Quebec, Canada

    Dirk Schlimm

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Godard, R. (2017). The Convolution as a Mathematical Object. In: Zack, M., Schlimm, D. (eds) Research in History and Philosophy of Mathematics. CSHPM 2016. Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne d’Histoire et de Philosophie des Mathématiques. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-64551-3_13

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