Who first introduced Generalized Functions?

Question

I've been looking online but I can't seem to track down the original paper introducing generalized functions. I suspect it should be by Schwartz (because of the Schwartz distributions) but I may definitely be wrong.

Answer 1

Generalized functions were first introduced at the end of the 1920-s by P.A.M. Dirac in his research on quantum mechanics, in which he made systematic use of the concept of the $\delta$-function and its derivatives.

Reference: https://www.encyclopediaofmath.org/index.php/Generalized_function

Answer 2

Wikipedia article has a pretty informative section on the history of Dirac delta. To summarize:

• The idea itself can be traced to the times of Fourier: since $$f(x)=\frac{1}{2\pi}\int\limits_{-\infty}^{+\infty}e^{ipx}\left(\int\limits_{-\infty}^{+\infty}e^{-ipq}f(q)dq\right)dp$$ it is natural to (non-rigorously) rearrange as $$f(x) = \int\limits_{-\infty}^{+\infty}\left(\frac{1}{2\pi}\int\limits_{-\infty}^{+\infty}e^{ipx}e^{-ipq}dp\right)f(q)dq = \int\limits_{-\infty}^{+\infty}\delta(x-q)f(q)dq$$ where $$\delta(x-q)=\frac{1}{2\pi}\int\limits_{-\infty}^{+\infty}e^{ip(x-q)}dp$$ Though, it seems that no one actually expressed $\delta$ as a separate notion.
• Throughout the 19th century, many authors have explicitly used the idea of a unit impulse in PDE theory, either through Fourier transform as above, as a limit of shrinking Gaussians, or by some other means. This is intimately connected to the notion of Green's function.
• The term delta function was coined by Dirac in his book The Principles of Quantum Mechanics that was published in 1930. He seems to have used this just as a notational convenience. The name delta comes from resemblance to Kronecker delta.
• The modern form of generalized functions, i.e. distributions defined as linear functionals on spaces of appropriate test functions, originate in works of Sobolev, later developed and extended by Schwartz in 1940s.