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What would be wrong with defining the Dirac delta function as $$ \int_{-\infty}^{\infty} \delta^{(n)}(f(x)) g(x) \,dx := \lim_{h\rightarrow 0}\int_{-\infty}^{\infty} \delta_h^{(n)}(f(x)) g(x) \,dx $$ for a suitable nascent delta function $\delta_h(x)$? For example, the rectangular pulse, hat function, and normal distribution nascent delta functions are all suitable in the case $n=0.$ For $n\leq1,$ the hat function and normal distribution $\delta_h$'s are suitable. For $n\geq 2,$ the normal distribution $\delta_h$ is suitable.

If the Dirac delta function is defined this way, it satisfies all the familiar properties of the Dirac delta function (the sifting property, composition rule, derivatives rule). So why is distribution theory necessary? Is the definition given above sufficient for most users of the Dirac delta function?

J. Heller
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    There are a lot more distributions than just the Dirac delta. Also, a lot of the machinery will be quite tedious to develop this way; for example, the derivative rule will be annoying to prove in this case, and you will have to do it separately for each distribution, even though the rule is universal. – Ian Jul 31 '16 at 18:37
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    Distribution theory is a lot more than the Dirac delta function. –  Jul 31 '16 at 18:37
  • @Ian Integration by parts could be used to easily prove the derivatives rule for every smooth nascent delta function, because the boundary terms vanish in the limit. – J. Heller Jul 31 '16 at 18:44
  • For the Dirac delta sure. But you have to do that over again for every distribution you work with. And you have to work with smooth approximants too, or else prove the derivative rule again for your nonsmooth approximants. It's easier to just set up distribution theory (there's really not that much there if you ignore the topology). – Ian Jul 31 '16 at 18:57
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    One must know also that mathematical distribution theory, based on duality (mainly elaborated by Laurent Schwartz in the years 1945-1950), though rather abstract is a formidable gardrail for the treatment in a thorough way of difficult issues. Nevertheless, "physicist's view" (as inaugurated by P. Dirac in the 1930s with a certain number of "rules of thumb" keeps its value for coping with not-too-difficult issues, without resorting to sometimes non-intuitive aspects of the mathematical "machinery". One must also know that there are challengers to Schwartz approach like that of Mikusinski. – Jean Marie Jul 31 '16 at 19:00
  • To clarify: I don't doubt that distribution theory is necessary for some things. I am only questioning whether it is necessary for most users of the Dirac delta function. – J. Heller Jul 31 '16 at 19:00
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    Necessary, no, but I routinely find people using the Dirac delta function making simple mistakes because their intuition does not quite match the exact behavior. They also don't really understand what doesn't work properly, for example that $\int_0^\infty \delta(x) dx$ is badly defined or that distributions can't be multiplied. – Ian Jul 31 '16 at 19:46

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