Understanding Dirichlet conditions

[spaghetti3451]

Could you please check if these are the Dirichlet conditions?

1. f(x) is periodic.
2. f(x) has a finite number of maxima and minima over one period.
3. f(x) is single valued, except at a finite number of discontinuities over one period.
4. \int^{-L/2}_{+L/2} \left|f(x)\right| dx is finite over some range of L.

These conditions are sufficient but not necessary, e.g. sin(1/x) violates condition 2.

 

[henry_m]

1 and 2 are correct.

3: A function is single valued by the definition of a function. This should read "f(x) is continuous, except at a finite number of discontinuities over one period."
4. This should specify that L is the period: f must be absolutely integral over a period.

There should also be an extra condition that f is bounded.

Then, these conditions are sufficient for the Fourier series to converge pointwise to f except at discontinuities. They are, however, not necessary, and many other conditions for convergence exist. For example, if f is differentiable at x, its Fourier series at x converges to f(x). Take f(x)=sin(1/x) for 0<x<L, and let f(x)=f(x+L) define f outside this range (with f(0)=0, say). Its Fourier series will converge everywhere except 0, despite the fact that condition 2 is broken.

 

[spaghetti3451]

One of the best replies on Physicsforums.

Thanks!