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!

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.

Hi Chiro!

Thanks for the reply.
Are you referring to to Fourier Transform or the Fourier Series? What you've described sounds a lot like the Series... Which I understand. I'm asking about the Transform, which Ive been led to believe, perhaps mistakenly, is not exactly a projection on an orthogonal basis.
Thanks!

Ohh ok that is a bit different.

I'm not too familiar with the integral transforms but there is a rich set of ideas associated with general transforms in the area known as Integral Transforms where the focus is on the kernel and the inverse kernel and what makes a particular kernel useful, which you might want to check out if you are keen on learning about the magic of these transforms.