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 DN 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 eit = 1 if and only if t is an integer multiple of 2π. Hence
The second follow by multiplying and dividing the second expression by eit/2, together with 2i sinz = eiz - 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.