I just started to read the book 'A Brief History of Time' by Stephen Hawking. Actually When he was talking of the idea of infinite density 'thing' before big bang suddenly the mathematical function came to my mind was delta Dirac impulse function I am familiar at engineering classes. But that was normally either in time domain or frequency domain and also it was one dimensional. The Fourier transform dual of delta function is flat constant right from minus infinity to plus infinity . How can I model a 2D and 3D delta Dirac function in space and find its corresponding Fourier transform dual. Also what will be dual of space in the Fourier domain?


Multidimensional Dirac delta is usually defined as $$\delta_{3D}(x,y,z)=\delta(x)\delta(y)\delta(z)$$

You can find its Fourier transform as a convolution of constants, which will still appear constant. Alternatively, you can just use the definition and find triple integral, which will give the same result:

$$\mathcal{F}\{\delta_{3D}(x,y,z)\}=\frac1{\left(\sqrt{2\pi}\right)^3}\iiint_{\mathbb R^3} e^{-i (k_xx+k_yy+k_zz)}\delta(x)\delta(y)\delta(z)\operatorname dV=\\ =\frac1{\left(\sqrt{2\pi}\right)^3}$$

Fourier-dual of space of positions $\vec r$ would be space of spatial frequencies, or wavevectors $\vec k$.


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