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The expression $\int_{-\infty}^\infty e^{i k x} dx$ is sometimes identified as the Dirac delta function. This identification is said "formal" or "symbolic", and some physics texts say that the theory of distribution makes this identification rigorous. So I read some texts on distributions and I think now I understand what delta function is as a distribution. However, I still cannot give any meaninig on the expression as a distribution. Is it possible to give a well defined meaning as a mathematical object to the expression?

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  • $\begingroup$ I'm not sure but I believe: its the inverse Fourier transform of 1, but of course writing it formally as this integral doesn't make sense, so you define as a limit of an approximating sequence in $\mathcal D'$. Check en.wikipedia.org/wiki/Oscillatory_integral $\endgroup$
    – Calvin Khor
    Aug 8, 2018 at 15:46
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    $\begingroup$ @CalvinKhor So you introduce two ways. i) The actually "intended" by the expression may be the distributional Fourier transform of 1. ii) Intepreting the improper integral as a special kind of limit introduced in the Wikipedia page makes the expression rigorous. Am I right? $\endgroup$
    – lyrin
    Aug 8, 2018 at 16:35
  • $\begingroup$ That's my impression but I'm not certain as I've only read about this in passing. $\endgroup$
    – Calvin Khor
    Aug 8, 2018 at 16:36

2 Answers 2

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The way to give meaning to an expression as a distribution is to see its effect on a test function. If $\phi(x)$ is a test function (a member of the Schwartz space $\mathscr S(\mathbb R)$), then the action of the formal expression $T = \int_{-\infty}^\infty e^{ikx}\; dx$ on $\phi$ is obtained by integration (and switching the order of the integrals): $$ T(\phi) = \int_{-\infty}^\infty dx \int_{-\infty}^\infty dk \; \phi(k) e^{ikx} = \int_{-\infty}^\infty dx \; \hat{\phi}(-x/(2\pi)) = 2 \pi \phi(0)$$

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  • $\begingroup$ Isn't it still "formal"? There is a distribtion $T$ defined as $\left< T, \phi \right> = \int_{-\infty}^\infty dx \int_{-\infty}^\infty dk \phi(k) e^{ikx}$. Do you mean I have to identify any occurrence of $\int_{-\infty}^\infty e^{ikx} dx$ in an integral with the action of $T$ even if the integral itself is not well defined? For example, $\int_{-\infty}^\infty dk \int_{-\infty}^\infty dx \phi(k) e^{ikx}$ is sometimes found in physics texts but it is meaningless if interpreted as it is. $\endgroup$
    – lyrin
    Aug 8, 2018 at 16:07
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$$ \langle \int_{-R}^{R}e^{ikx}dk,\varphi\rangle=\int_{-\infty}^{\infty}\varphi(x)\int_{-R}^{R}e^{ikx}dkdx \\ = \int_{-R}^{R}\int_{-\infty}^{\infty}\varphi(x)e^{ikx}dx dk $$ As $R\rightarrow\infty$ the above tends to $2\pi\varphi(0)$. So the truncated integral converges in the distribution sense to a scalar times the Delta function at $0$: $$ \int_{-R}^{R}e^{ikx}dk \rightarrow 2\pi\delta_{0}. $$ The improper integral converges in the dual. It's an element of $\mathscr{S}^*$.

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