I'm just beginning learning about Fourier transforms and if I look up the Fourier transform for a function, say $\cos(w t)$, I find results with multiple different coefficients. I've seen three so far: $\pi$, $\sqrt{\frac\pi2}$, and $\frac12$, each multiplied by $[\delta(w-w_0)+\delta(w+w_0)]$. Are these equivalent and if so how? How do you know which to use?
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The coefficient of the transformed function depends on your definition of the Fourier transform and the inverse transform. For example, I could define the forward transform and inverse respectively as
$$F(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(x) e^{-ikx}\ dx$$ $$f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty F(k) e^{ikx}\ dx$$
But this needn't be your definition. Wikipedia has a nice table of common definitions for the Fourier transform. They're all...fairly similar. The one I chose looks like their second one in that table (where $n=1$). We use the FT to move a problem from one space to another, solve it there, and then transform it back. That's the point - to change spaces to another one without losing any information - and then to recover that information.
Now the point you bring up is how can all these different transforms be equal? Well, they aren't equal if you choose any random inverse transform. That's not how it works. Think of the forward and inverse transform as a matching set of keys. One, letting you from one space to the other, and the other reopening the same door in the reverse direction.
That being said, if you choose the matching inverse transform from your definition, then you will transform back to whatever space you began with, recovering all information.
End point? Pick a definition and stick to it.
zahbaz
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The Fourier transform has different definitions in different settings. Some have a $\frac{1}{\sqrt{2\pi}}$, some have it without the square root, some have no such factor. The inverse transform is also defined with these $2\pi$ factors so that no matter which convention you use a consistent result is obtained. Partly it is a question of units, i.e. whether you want the Fourier Transform in terms of angular or simple frequency. So it is purely a question of convention.
Paul
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It's hard to wrap my head around this. How can you start with the same function, get different amplitude results, and then say it's only convention? I mean, I haven't got to the point where I'm actually doing anything real with it, but I would think that in a real scenario the amplitude would be important and not just chosen arbitrarily? – Austin Nov 23 '15 at 00:19
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Hmm the angular vs simple frequency makes sense and I guess that accounts for 2/3 of the results I've found. – Austin Nov 23 '15 at 00:28
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@AustinMW, the multiplicative factor doesn't change the overall results. For example, if our multiplied by a number for a transform, and you divided by the reciprocal of the number for the inverse, it doesn't matter if that number is 2 or 5. In the end, you're going to get to the same result. – Paul Nov 23 '15 at 00:32
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Check out the Wikipedia page:
There are several common conventions for defining the Fourier transform ...
Wolfram Mathworld has at least one alternate definition.
The problem is that different branches of mathematics and physics use different definitions for the Fourier Transform. Each definition gives a slightly different result.
So perhaps we might given them different names, but they all tend to have the same (essentially) properties, so the fields just pick the constants that makes it work best for their purpose, leaving confusion in their wake.
Thomas Andrews
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