As you know Fourier Tansform bounds are from to and for sine and cosine transforms is from 0 to . How can apply Fourier Transform to a finite domain? I mean by some kind of transform function!
If the transformations are different for different Fourier transforms, Please give me all.
2. For any complete ortho-normal basis for an appropriate class of functions on (with respect to some weight if we want) we have:
Now the basis functions in you post are certainly orthogonal, and with are orthonormal on , so we need only show that they are a complete basis (which I don't have time to do at present, I may come back to this).