# Origin of the convolution theorem

I am a chemist, with some interest in signal processing. Sometimes, we use the deconvolution process to remove the instruments response from the desired signals. I am looking for the earliest reference which proposed the convolution theorem which is often utilized in signal processing (i.e., convolution becomes a multiplication in the Fourier domain).

The Earliest Known Uses of the Word of Mathematics websites gives lot of details on the word convolution, but who was the first person to specifically show the above mentioned property- the connection of Fourier transforms with convolution?

Here is the history of Convolution Operation: https://pulse.embs.org/january-2015/history-convolution-operation/

However, a mathematician privately disagreed with that historical account given in this article.

Thanks.

The general theorem that Fourier’s $$\mathscr F:L^2(G)\to L^2(\hat G)$$ maps convolution to product and vice versa is in Weil (1940, p. 113), for any locally compact abelian $$G$$ with dual $$\smash{\hat G}$$. But the special cases of the real line, the circle group (with $$(f,g,h)(t)=$$ $$\sum (a_n,b_n,c_n)e^{int}$$), and the integers: \begin{align} &\mathbf{(R)}\qquad h(t)=\textstyle\int_{-\infty}^\infty f(s)g(t-s)\,ds &&\Rightarrow &&\textstyle\mathscr F(h)=\mathscr F(f)\mathscr F(g)\\ &\mathbf{(T)}\qquad h(t)=\smash[t]{\textstyle\int_0^{2\pi} f(s)g(t-s)\,ds} &&\Rightarrow &&c_n=a_nb_n\\ &\mathbf{(Z)}\qquad h(t)=f(t)g(t) &&\Rightarrow &&\textstyle c_n=\sum_k a_kb_{n-k}\\ \end{align} were known earlier:

• Dieudonné (1981, p. 195) and Mackey (1980, p. 628) attribute $$\mathbf{(R)}$$ to Lyapunov (1900, 1901?); it’s also in Hausdorff (1901, p. 169), Poincaré (1912, p. 207), Lévy (1925, p. 184; 1928, p. 79), etc. (They state results in terms of added independent random variables, not repeating the step that $$h(t)dt$$ is the image of the product measure $$f(s)ds\times g(t)dt$$ under addition $$\mathbf{R\times R\to R}$$.)

• Kahane and Lemarié-Rieusset (1995, p. 19) attribute $$\mathbf{(T)}$$ to Fourier (1822, p. 259) where it is rather implicit; it’s explicit in Young (1912, p. 32).

• Burkhardt (1901, p. 84; 1914, p. 947) finds $$\mathbf{(Z)}$$ in Cauchy (1844, p. 1125) and earlier in Euler (1760), for both trigonometric polynomials (“product-to-sum”, pp. 176-186) and some series (p. 200); it’s also in Pringsheim (1886, p. 158), Hurwitz (1902, p. 369), Lebesgue (1906, §52), etc.

(And indeed, the history in Domínguez (2015, p. 46) cheerfully ignores all of the above.)

Added: Adams (2009) contains translations of Lyapunov’s (1900, 1901). This should be consulted, as e.g. Fischer (2011, p. 201) contradicts Dieudonné and Mackey by saying: “Lyapunov never used general concepts such as inversion formula or correspondence between convolution of distributions and products of characteristic functions” (and I am indeed not really seeing $$\mathbf{(R)}$$ in (1900)).

• Since this seems to be a better answer than the one given to hsm.stackexchange.com/q/9762/4490 (which relied on Domínguez), maybe it should be linked from there? I'm uncertain of what the proper procedure is. Jun 28, 2019 at 12:48