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

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

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

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

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 is stated and used to invert the convolution:

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

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