4
$\begingroup$

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:

Paper image

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

Book

and the note 200 reads:

Book

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.

$\endgroup$
5
  • 1
    $\begingroup$ I would look at the history of Green (Green's) functions, the Heaviside operational calculus and associated Bromwich-Laplace transform and fractional.calculus, and integral transforms as discussed by Titchmarsh and Hardy for correlates of deconvolution, particularly where the Green/influence function or integral kernel is of the form G(x-y). $\endgroup$ Mar 28, 2020 at 15:58
  • 1
    $\begingroup$ @M.Farooq --- Pincherle's 1886 paper is online here --- it does not seem to address Fourier transformations or convolutions, at least not in any explicit way (did I overlook it?) $\endgroup$ Mar 30, 2020 at 13:32
  • 1
    $\begingroup$ Thanks Dr. Carlo. I think the reference by G. Doetsch, Theorie und Anwendung der Laplace-Transformation is not correct. If we see the Table 1, here pulse.embs.org/january-2015/history-convolution-operation, it mentions a 1907 reference by Pincherle. No citation is provided there. $\endgroup$
    – AChem
    Mar 30, 2020 at 13:58
  • 2
    $\begingroup$ indeed, I think I have identified this 1907 reference; I copy the relevant text in the answer box. $\endgroup$ Mar 30, 2020 at 14:36
  • $\begingroup$ By the way, I think that there is a common tendency in mathematics history to attribute 'generously', identifying the first place where recognisable precursors of an idea appear, not just the first place it appears in its modern form. (Fourier would probably not recognise what we now call the Fourier transform.) Quite possibly that's the spirit of the reference to Maréchal and Croce, although I am no expert on the physics and so am not confident I'd recognise the idea in the paper even if it were there …. $\endgroup$
    – LSpice
    Mar 30, 2020 at 14:47

2 Answers 2

3
$\begingroup$

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.)


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] $\gamma(u)/\alpha(u)$.

$\endgroup$
7
  • $\begingroup$ I am also wondering that mathematicians must be using this a little bit earlier given that convolution was known in 1900s? Why there is a delay of 50 years for undoing convolution? What do you think? $\endgroup$
    – AChem
    Mar 27, 2020 at 16:08
  • $\begingroup$ Thank you very much! I think that is it. $\endgroup$
    – AChem
    Mar 30, 2020 at 14:46
  • $\begingroup$ Thanks for the link to the article by Pincherle. Have you access to Heaviside's Electromagnetic Theory Vol. 3? Supposedly the section "The solution of definite integrals by differential transformation" has some presentation of the inverse Laplace/Bromwich/Fourier-Mellin transform, which Heaviside discussed with Bromwich. $\endgroup$ Mar 31, 2020 at 2:43
  • 2
    $\begingroup$ @TomCopeland --- Heavide's book is online here --- I looked at the section you mention but could not find anything hinting at deconvolution. $\endgroup$ Mar 31, 2020 at 6:44
  • 1
    $\begingroup$ Thanks. Note that eqn. 11 on p. 236 is a statement of the inverse Laplace transform of $1/p^{n+1}.$ Heaviside is taking the Laplace transform of $\int_0^{\infty} e^{-px} (x^n/n!)dx$ with $p = d/dx = D$. Then, with $H(x$) the Heaviside step fct., evaluating $D^{-(n+1)}H(x)= H(x)x^{n+1}/(n+1)!$, which is the inverse Laplace trf of $1/p^{n+1}$. Bromwich felt this method generalized by Heaviside was superior to the use of the inverse Laplace transform as a complex contour integral, which Bromwich was the first to present, I believe. $\endgroup$ Mar 31, 2020 at 17:19
4
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ Thanks, Unfortunately, the original paper is not available anywhere. I checked HathiTrust and Internet Archive. Hathi Trust has all copies of Sitzungsberichte (1931) but it is locked due to copyright even from viewing a single page. $\endgroup$
    – AChem
    Mar 29, 2020 at 19:05
  • 1
    $\begingroup$ A close connection between the symbolic calculus and FT deconvolution methods is illustrated by Bracewell's derivation of Eddigton's formula for correction of optical spectral line broadening due to limited resolution of the spectrometer. See the 1913 article by Eddington (academic.oup.com/mnras/article/73/5/359/972797) and the section titled Eddington's Formula in Bracewell's book Fourier Analysis and Imaging. Very likely Bracewell used FT deconvolution when he did R&D on radars and imaging by radiotelescopes before 1954, well before he wrote his books on the FT. $\endgroup$ Mar 30, 2020 at 6:08
  • $\begingroup$ Thanks Tom, I have updated the post. Your pointer to Pincherle as the earliest one is the right version as it can be seen in the early post. $\endgroup$
    – AChem
    Mar 30, 2020 at 13:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.