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Chapter 5 / Section 5.1, 5.2 - ok to consider delta distribution?
« on: March 09, 2020, 10:49:21 PM »
From a physics class, I remember the statement "the Fourier Transform of a plane wave is the delta distribution", e.g. for the unitary transform, $\widehat{e^{ibx}} = \sqrt{2 \pi}\ \delta (k - b)$. I understand that these plane waves are not "square integrable", so we do not formally consider them. But I wonder if it is acceptable to use this on homework / quiz / test?
For example, when trying to calculate the transform of a function like $f(x) = e^{-\alpha \left|x\right|}\cos \beta x$, I want to consider the exponential and the $\cos$ functions separately, find their Fourier Transforms, and then convolute. The $\cos$ will give these delta distributions by Euler's Identity, and the convolution will be easy.
Cheers,
Jack
For example, when trying to calculate the transform of a function like $f(x) = e^{-\alpha \left|x\right|}\cos \beta x$, I want to consider the exponential and the $\cos$ functions separately, find their Fourier Transforms, and then convolute. The $\cos$ will give these delta distributions by Euler's Identity, and the convolution will be easy.
Cheers,
Jack