[PDF] application of fourier series in mathematics

What is a Fourier series?

A Fourier series is an expansion of a periodic function f (x) in terms of an infinite sum of sines and cosines. Fourier Series makes use of the orthogonality relationships of the sine and cosine functions. A difficult thing to understand and/or motivate is the fact that arbitrary periodic functions have Fourier series representations.

What is the best application of Fourier analysis?

Signal Processing. It may be the best application of Fourier analysis. Approximation Theory. We use Fourier series to write a function as a trigonometric polynomial. Control Theory. The Fourier series of functions in the differential equation often gives some prediction about the behavior of the solution of differential equation.

Can Fourier series be taken for a truncated version of a function?

Fourier series can be taken for a truncated version of the function or to the periodic summation . Frequently when describing how Fourier series behave, authors introduce the partial sum operator for a function . (Eq.

Which symmetry properties of functions are useful in the study of Fourier series?

There are two symmetry properties of functions that will be useful in the study of Fourier series. A function f (x) is said to be even if f (?x) = f (x). The function f (x) is said to be odd if f (?x) = ?f (x). Graphically, even functions have symmetry about the y-axis, whereas odd functions have symmetry around the origin.