Suppose you have a reasonable continuous function on some interval, say on , and you want to approximate it by a trigonometric polynomial. A straightforward approach is to write

where and are the Fourier coefficients:

(Integration can be done with the standard Calculus torture device). With , we get

which, frankly, is not a very good deal for the price.

Still using the standard Fourier expansion formulas, one can improve approximation by shifting the function to and expanding it into the **cosine** Fourier series.

where

Then replace with to shift the interval back. With , the partial sum is

which gives a much better approximation with **fewer** coefficients to calculate.

To see what is going on, one has to look beyond the interval on which is defined. The first series actually approximates the periodic extension of , which is discontinuous because the endpoint values are not equal:

Cosines, being even, approximate the symmetric periodic extension of , which is continuous whenever is.

Discontinuities hurt the quality of Fourier approximation more than the lack of smoothness does.

Just for laughs I included the pure sine approximation, also with .