Conditions for Fourier series

Suppose a function f(x) has a period of 2π and is integrable in a period [-π, π]. Now there are two conditions.

The function f(x) with period 2π is absolutely integrable on [-π, π] so that the following Dirichlet integral of this function is finite:
[Tex]\int\limits_{ – \pi }^\pi  {\left| {f\left( x \right)} \right|dx}  \lt \infty ; [/Tex]

Next condition is that the function is a single valued, piecewise continuous (must have a finite number of jump discontinuities), and piecewise monotonic (must have a finite number of maxima and minima).

If conditions 1 and 2 are satisfied, the Fourier series for the function exists and converges to the given function. This means that the sum of the Fourier series of any given function converges back to give the same function. This is the basic definition of the Fourier series expansion. Before further understanding the concept of the Fourier Series we should first understand the concept of odd and even functions and periodic functions.

• Odd function: Suppose we are given a function y = f(x).   

f(-x) = -f(x) = -y

then the function is said to be odd.

• Even function: Again consider a function f(x) = y.

If f(-x) = f(x) = y

Then the function is even in nature.

• Periodic functions: Let a function f(x) be periodic with an interval λ. Now consider an element x as a part of the domain of this function. This means that,

f(x) = f(x + λ)

Hence periodic functions are those functions that repeat themselves over an interval of values(λ as shown above). The smallest possible positive value of λ is called the period of this function.

Fourier Series is a sum of sine and cosine waves that represents a periodic function. Each wave in the sum, or harmonic, has a frequency that is an integral multiple of the periodic function’s fundamental frequency. Harmonic analysis may be used to identify the phase and amplitude of each harmonic. A Fourier series might have an unlimited number of harmonics. Summing some, but not all, of the harmonics in a function’s Fourier series, yields an approximation to that function. For example, a square wave can be approximated by utilizing the first few harmonics of the Fourier series.

In this article, we will learn about Fourier Series, Fourier Series Formula, Fourier Series Examples, and others in detail.