[Math] Integral definition of delta function and Kronecker symbol

dirac deltadistribution-theorykronecker-symbol

I know the following two definitions for the delta function and Kronecker delta, respectively:

(1) $\int_{-\infty}^{\infty}\frac{e^{iwt}}{2\pi}\mathrm{d}t = \delta(w)$

(2) $\int_{-\pi}^{\pi}\frac{e^{i(m-n)t}}{2\pi}\mathrm{d}t=\delta_{m,n}$

First question: What is the difference between the delta function and the Kronecker delta? Especially in view of eq. (1), couldn't I say (2) $=\delta(m-n)$? Or would this mean something different?

Second question: Why are the integration limits different? I would think that I could replace $w$ in (1) by $(m-n)$ to get to definition (2), but obviously this wouldn't work because of the different integration limits.

Best Answer

You presented two formulas for Kronecker delta and Dirac delta, neither or which are really used as definitions of those concepts. In practical terms, a remedy for confusion is to read the actual definitions of Dirac delta and Kronecker delta and do some work with these things.

But yes, there is a relation here. 

+------------+--------------------+-------------------+
|            |  Kronecker delta   |    Dirac delta    |
+------------+--------------------+-------------------+
| Arguments  | Integers           | Real numbers      |
| Dual space | Circle             | Line              |
| Functions  | Periodic functions | L^2 functions     |
| Expansion  | Fourier series     | Fourier transform |
+------------+--------------------+-------------------+

The duality referred to is Pontryagin duality. It leads from integers to a circle (which we represent by $[-\pi,\pi]$), to periodic functions, to Fourier series. The same line of reasoning leads from real numbers to $L^2$ functions to Fourier transform.

The formulas you stated describe the Fourier series/transform of Kronecker/Dirac delta. Despite the many parallels between the series and transform, one does not get from one to another by mere formal manipulations. See, e.g., Difference between Fourier series and Fourier transformation

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