Fourier Transform of $1$

Question

I was wondering, why is the Fourier Transform of $1$ equal to the Dirac Delta Function? I found derivations based on the inverse Fourier Transform, but can someone show the Fourier Transform integral converges to this specific function? Also is the Dirac Delta Function that is the result of this transform equal to $1$ or infinity at $0$? I ask because I've found conflicting results in literature.

Answer 1

Assuming the Fourier transform of $f(t)$ is defined as

$$\mathcal{F}_t[f(t)](x)=\int\limits_{-\infty}^{\infty} f(t)\,e^{-2 i \pi x t}\,dt\tag{1}$$

the Fourier transform of $f(t)=1$ is

$$\mathcal{F}_t[1](x)=\int\limits_{-\infty}^{\infty} e^{-2 i \pi x t}\,dt=\delta(x)\tag{2}$$

which leads to the limit representations

$$\delta(x)=\underset{f\to\infty}{\text{lim}}\left(\int_{-f}^f e^{-2 i \pi x t}\,dt\right)=\underset{f\to\infty}{\text{lim}}\left(2 f\, \text{sinc}(2 \pi f x)\right)\tag{3}$$

and

$$\theta(x)=\int_{-\infty}^x \delta(y)\,dy=\underset{f\to\infty}{\text{lim}}\left(\frac{1}{2}+\frac{\text{Si}(2 \pi f x)}{\pi}\right).\tag{4}$$

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The following figure illustrates formula (3) for $\delta(x)$ above evaluated at $f=4$.

Figure (1): Illustration of formula (3) for $\delta(x)$ evaluated at $f=4$

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The Dirac delta function $\delta(x)$ is sometimes left undefined at $x=0$ (Mathematica evaluates $\delta(x)$ to $0$ for real $x\ne 0$ but refuses to evaluate $\delta(0)$), but using formula (3) for $\delta(x)$ above leads to

$$\delta(0)=\underset{f\to\infty}{\text{lim}}(2 f\, \text{sinc}(2 \pi f\, 0))=\infty\,.\tag{5}$$

since $\text{sinc}(0)=1$.

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The following figure illustrates formula (4) for $\theta(x)$ above evaluated at $f=4$.

Figure (2): Illustration of formula (4) for $\theta(x)$ evaluated at $f=4$

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Also note that using formula (3) for $\delta(x)$ above leads to

$$\int_{-\infty}^{\infty} \delta(x)\,dx=\int_{-\infty}^{\infty} 2 f\, \text{sinc}(2 \pi f x)\,dx=1\tag{6}$$

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This answer user reuns posted to one of my questions on Math StackExchange provides a more rigorous answer with respect to the Fourier transform of $1$.