What is it called when you replace sine with cosine in a Fourier series?

Question

Suppose you have a Fourier sine series:

$$f(t) = \sum_{n=0}^{\infty} a_n \sin(n \omega t)$$

and you replace sine with cosine:

$$g(t) = \sum_{n=0}^{\infty} a_n \cos(n \omega t)$$

or conversely, replace cosine with sine. Is there a name for what $g$ is called with respect to $f$?

Also, when you take a square wave:

$$f(t) = \frac{4}{\pi} \sum_{n=1}^{\infty} \frac{\sin((2n - 1)\omega t)}{2n - 1}$$

and make the corresponding $g$:

$$g(t) = \frac{4}{\pi} \sum_{n=1}^{\infty} \frac{\cos((2n - 1)\omega t)}{2n - 1}$$

what is what you get called?

Answer 1

It is called the Hilbert transform. Look at 15.4.1 in http://dsp-book.narod.ru/HFTSP/8579ch15.pdf

Actually, changing $\sin$ to $\cos$ is minus the Hilbert transform. But changing $\cos$ to $\sin$ is the Hilbert transform.

http://en.wikipedia.org/wiki/Hilbert_transform#Table_of_selected_Hilbert_transforms

Answer 2

The Fourier sine series of f is obtained for an odd function while the Fourier cosine series of f is obtained for an even function. I do not see any relation between those two and I do not think that, as copper hat said, that there is any particular name for this.