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As a physicist, I have frequently encountered the argument that a function may be approximated as a sum of delta functions. However, the exact correspondence is never given. What is it?

My attempt yields the following equation:

$$f(x) \approx f_{\text{approx}}(x) \equiv \sum_n \delta(x-x_n) \, \Delta x \, f(x_n)$$

where $\Delta x$ is understood to be a finite real number approaching zero and $x_n \equiv n \, \Delta x$. I found this by requiring that a definite integral of $f_{\text{approx}}$ over a given interval should tend toward the definite integral of $f$ over that same integral.

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Formally we have $\displaystyle f(x)=\int_{-\infty}^\infty f(y)\delta(x-y)\,dy\,,$ and approximating this integral by its corresponding Riemann sums gives what you wrote, in a physicist's sense.

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  • $\begingroup$ As a physicist I am compelled to ask what you mean by "in a physicist's sense". $\endgroup$ – DanielSank Dec 27 '14 at 2:28
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    $\begingroup$ Well, what I meant is that it needs to be interpreted properly to make sense mathematically, and it seems it often isn't made precise in physics classes. $\endgroup$ – JLA Dec 27 '14 at 3:31
  • $\begingroup$ Ah, yes indeed. Let us not feel obligated to qualify answers based on established failures of physics pedagogy. (that's a joke) $\endgroup$ – DanielSank Dec 27 '14 at 3:54

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