**Branch :**Computer Science and Engineering

**Subject :**Math -2

## Convergence of Fourier Series

**Convergence of Fourier Series:**

For each positive integer N, let

This function is continuous and periodic with periods 2π. Note also that

The function D_{N} is called the** Dirichlet Kernel.**

**Preposition 2 **If t is not an integer multiple of 2π, then

**Proof **: To obtain the first equality, note that e^{it} = 1 if and only if t is an integer multiple of 2π. Hence

The second follow by multiplying and dividing the second expression by e^{it/2}, together with 2i sinz = e^{iz} - e^{-iz}

**Proposition 3** If f is integrable, then

A change of variables x - t = s finishes the proof.

We say that a function f : [0, 2π] → C satiesfies a Lipschitz condition if there is positive constant M such that the periodic extension of f-satiesfies

In terms of the functions f defined in [0, 2π] this can be expressed as;

for all s, t ε [0, 2π] . The min is the distance from s to t modulo 2π, and so it is the distance as measured on a circle of length 2π.