# How to intuitively understand the Fourier Transform

• Jan 28th 2013, 04:16 PM
sapz
How to intuitively understand the Fourier Transform
Hello!

I've recently learned about the Fourier Series from the point of view of projection onto the Space spanned by the orthogonal functions 1,cos(x),sin(x),cos(2x),...

And so when I think about the Coefficients of the Fourier series, I think about how "much" of the function exists in that particular frequency. How "much" the cos(kx) or sin(kx) frequency constitutes the original function.

So now Im trying to learn about the Transform. The technique itself is of less importance, I'd like to gain an intuitive understanding about what it means... And so far all I could find is more about the technique.

Could you try to connect the dots for me, and tell me what resemblence does the transform have with the series? If you could paint in the same colors Ive mentioned about my current understanding of the Fourier Series that would be fantastic...

Thank you!
• Jan 28th 2013, 05:38 PM
chiro
Re: How to intuitively understand the Fourier Transform
Hey sapz.

Basically what this is, is an example of treating a function like a vector (in fact, it's an infinite dimensional vector space as opposed to the normal vectors which are in a finite dimensional space).

What you are doing is taking a function and treating it like an infinite-dimensional vector and breaking it down by projecting it to a set of orthogonal vectors to get the component of the function that belongs to a particular vector.

Since all the vectors in the fourier basis are orthogonal to each other, it means that each projection will contribute only to that basis vector and no others (just like i,j,k are all orthogonal in R^3).

So what you are doing is in fact treating a function like a vector and projecting to a particular basis (namely the Fourier basis).

There are in fact, many different bases that you can project to just like there are many different orthogonal bases in R^3 (not just i,j,k) and this particular kind of thing is the subject of Fourier Analysis and Orthogonal Polynomials and functions.
• Jan 28th 2013, 05:42 PM
sapz
Re: How to intuitively understand the Fourier Transform
Hi Chiro!