Review In Fourier series

It is known that Fourier series are one of linear transformations and have many uses in mathematics, physics, engineering…etc. It is used to influence on periodic functions that repeats in a certain period to transform it into a sum of sine and cosine functions also used for non-periodic functions, as we will notice, and it has many other uses. In this article, I study, with examples, the important uses of Fourier series in physics, engineering, and mathematics, with proofs and examples where it necessary. One of results that we have reached in this article is that use Fourier series to facilitated a solution of differential equations in particular partial differential equations by the sinusoidal part of it, also obtaining the best approximation element for a particular function by means of its mathematical mean which called Feger operators. Moreover, there are more than one auxiliary conclusion for, example if a function has two different periods, then the sum of these periods, its subtraction, and its multiplying by constant, will be a new period for this function. Here, I refer to another conclusion that, Fourier series itself do not give the best approximation element for a particular function.