Toronto Math Forum

APM346--2020S => APM346--Lectures and Home Assignments => Chapter 5 => Topic started by: jfarrellhfx on March 09, 2020, 10:49:21 PM

Title: Section 5.1, 5.2 - ok to consider delta distribution?
Post by: jfarrellhfx 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
Title: Re: Section 5.1, 5.2 - ok to consider delta distribution?
Post by: Victor Ivrii on March 10, 2020, 08:35:56 AM
Then you need to explain more or less rigorously, what is $\delta$-distribution and how Fourier transform is defined for it. Yes, online textbook covers this topic and much more than your physics class but in more advanced chapters, we do not cover