Fourier Series Research
By Ali Albaghdadi Id: 20180007205
History of Fourier Research:
Fourier series have made a huge impact on applied mathematics and theoretical. Fourier series
are the sum of the orthonormal families which can approximate periodic, complex valued
functions with arbitrary precision. There is a quite good number of methods for estimating
complex numbers, but Fourier series are especially stand out because uniform convergence of the
Fourier series. Furthermore, Fourier coefficients are made to minimize the error from the actual
function. In mid eighteenth century, physical problems such as the conduction of patterns of heat
and the study of oscillations and vibrations led to the study of Fourier series. The major issue that
was faced is how arbitrary real valued functions could be presented by sum of simpler functions.
Fourier series is an infinite sum of trigonometric functions that are used to model real live valued
periodic functions. It has been used in signal processing and digital image processing for the
analysis of a single image as a 2-D wave form, and many other types of similar quantum
mechanics. Not only that but it also represents filter transformation, representation and encoding
and much more.
The Fourier breaks a function of time or signal into frequencies that makes in a way to how a
musical chart can be broken into nodes and represented as pitches. It is also called the frequency
domain representation of the original notes. Let S(x) be a periodic function with period [
−π,π¿
a
n
=1
T
∫
−π
π
S
(
x
)
cos
(
2πx∗n
2T
)
dx , n ≥ 0
