Earliest use of deconvolution by Fourier transforms

Question

From a previous discussion here Origin of the convolution theorem, it was shown that the property of convolution $y(t)$=$a$*$b$ becoming a multiplication after Fourier transform: $F$$(y(t))$= $F(a)F(b)$, was well known by early 1900s and clearly mentioned in 1941.

I was searching the earliest known use of deconvolution by Fourier transforms. Surprisingly, the term deconvolution is quite recent as per the unabridged version of the Oxford English Dictionary (OED). In deconvolution, two functions are divided in the Fourier domain to recover the original function, say $a$, if $y(t)$ and $b(t)$ are known. For example, if we wish to recover $a$, we can divide $F(y(t))$ by $F(b)$ and do an inverse transform to get $a$. It may not be a rigorous way, but it is a popular technique in spectroscopy from an empirical perspective.

OED mentions a 1967 paper titled "Posteriori image-correcting "deconvolution" by holographic fourier-transform division" in Physics Letters. The authors show the following:

The authors cite Maréchal and Croce as the first example in Comptes Rendus, however Gallica Original Paper does not have a single equation and the word Fourier is mentioned only in the first two lines! So this reference seems to be incorrect.

I am not interested in image analysis, but rather in the earliest known use of division process in the Fourier domain to recover original functions.

a) I wanted to know if mathematicians were using this approach well before 1960s to recover an original function from given convolution?

b) Spectroscopists call the division in the Fourier domain as deconvolution, what do mathematician call the process of division of two functions in the Fourier domain?

Update (30 Mar 2020)

From the detailed response by Tom Copeland, and the Table 1 shown in History of Convolution one can see another reference from 1943,

G. Doetsch, Theorie und Anwendung der Laplace-Transformation. New York: Dover, 1943

and the note 200 reads:

The reference to Picherle is given as "I. Studi sopra alcune operazioni funzionali. Mem. Accad. Bologna (4) 7 (1886)."

However the Table 1 of the convolution history mentions 1907. No reference is provided.

Thanks.

Answer 1

An early use of division in Fourier space to undo a convolution is Fourier Treatment of Optical Processes (1952), by Peter Elias, David S. Grey, and David Z. Robinson. (This paper precedes the paper by Maréchal and Croce cited in the OP.)

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Following the OP's and Copeland's lead to Pincherle suggests this 1907 publication Sull'inversione degli integrali definiti. The convolution theorem for Laplace transforms [called "funzioni generatrici" – generating functions; "funzione determinante" is the inverse transform] is stated and used to invert the convolution by dividing the transformed functions:

_{From eq. (10), or the equivalent (10'), we immediately find the answer to question number 4. Indeed, equation (d) $$\frac{1}{2\pi i}\int a(x-t)f(t)dt=g(x)$$ is equivalent [for the Laplace transforms] to $Gg=Ga\cdot Gf$ or $\gamma(u)=\alpha(u)\phi(u)$, and hence the function $f$ is determined by [the inverse transform of] $\gamma(u)/\alpha(u)$.}

Answer 2

Early uses of deconvolution via integral transforms:

A) Signal processing:

$$ \int_{\infty}^{\infty} K(y-x) h(x)dx = \int_{-\infty}^{\infty} e^{-i 2 \pi \omega (y-x)} h(x)dx $$

$$= e^{i2 \pi \omega y}\hat{h}(\omega)=H(\omega)$$

is an example of a convolution.

Dephasing $H$ and taking an inverse FT amounts then to deconvolution of the type you designate :

$$\int_{-\infty}^{\infty} \frac{H(\omega)}{e^{i2 \pi \omega y}}e^{i2 \pi \omega x} d \omega= h(x).$$

This must have been done at least by the researchers, such as Schwinger, at the MIT Radiation Lab during WW II in the development of radar.

B) Deconvolution via Fourier transforms of the Wiener-Hopf integral equation published in 1931:

Lawrie and Abrahams present in "A brief historical perspective of the Wiener-Hopf technique", the solution developed by Wiener and Hopf of the convolutional equation

$$ \int_{0}^{\infty} k(x-y) f(y) dy =\left\{\begin{matrix} g(x), & x > 0\\ h(x), & x<0 \end{matrix}\right.$$

where $f(x)$ and $h(x)$ are unknown. For $h(x)=0$, the solution specializes to the inverse transform of a ratio of Fourier transforms

$$ FT[HV(x)g(x)]/ FT[k(x)] = FT[HV(x)f(x)]. $$

$ HV$ is the Heaviside step function.

(Norbert Wiener had a vast range of interests, and, since signal propagation/processing had recently become important due to the development of telegraphs, power lines, telephones, radar, and x-ray diffraction, it seems plausible that he was one of the earliest to publish on deconvolution through the Fourier transform. The Mellin and Laplace transforms and deconvolutions are better suited for development of the operational/algebraic calculus explored by Lebnitz, Euler, and dozens after them.)

C) Operational calculus, fractional calculus, differential algebra:

For Heaviside's operator calculus (and use by Dirac), see the discussion, references, and comments at Ron Doerfler's post at his website Dead Reckonings. (Synowiec is also cited below, and see this note by Davis on Bromwich's views of the Heaviside calc.)

For differential algebra in general, read "Some highlights in the development of algebraic analysis" by Synowiec in which symbolic methods, the Heaviside calc, and the Laplace transform are stressed, but Norbert Wiener's Fourier transform method is only briefly mentioned with a reference to his 1926 book On the Operational Calculus. Pincherle's contributions are mentioned as well as by Dominguez.

Quoting Dominguez (from his timeline table):

1907 Despite the many occurrences and uses of the CCO, none of the previous authors made a complete study of it. The earliest one is, perhaps, that made by the Austrian-born mathematician Salvatore Pincherle (1853–1936) in connection with the solution of the complex integral equation

$$ \frac{1}{2 \pi i} \int_{|z| = P} k(s-z) f(z) dz = g(s)$$

where $P > 0$ and $k(z)$ and $g(z)$ are given functions, while $f(z)$ is unknown. Pincherle succeeded in the solution of this CCO using as tool the Laplace transform. .... . These results are the basis for the deconvolution method established in [35].

Pincherle also developed an axiomatic approach to the fractional calculus (which can be based on the Mellin convolution). See "The Role of Salvatore Pincherle in the Development of Fractional Calculus" by Mainardi and Pagnini. The solution to the op eqn.

$$ D^r HV(x)f(x) = HV(x)g(x)$$

is $$HV(x)f(x) = D^{-r}HV(x)g(x) = D^{-r}D^rHV(x)f(x),$$

which can be expressed as a deconvolution.

From "Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives" by Hilfer, Luchko, and Tomovski:

In the 1950’s, Jan Mikusinski proposed a new approach to develop an operational calculus for the operator of differentiation .... This algebraic approach was based on the interpretation of the Laplace convolution as a multiplication in the ring of the continuous functions on the real half-axis. The Mikusinski operational calculus was successfully used in ordinary differential equations, integral equations, partial differential equations and in the theory of special functions.