Can a Fourier series converge if Dirichlet conditions aren't all satisfied?

[Benny]

I have a question about Fourier series that I would like some help with. If there is a function f(t) which does not satisfy all of Dirichlet's conditions then can its Fourier series still represent it? All I've got is that if all of Dirichlet's conditions are satisified by f(t) then the Fourier series converges to f.

There isn't anything which says that if not all of conditions are satisfied then the Fourier series cannot converge to the function f(t). So I'm having trouble drawing a conclusion. Can someone help me out? Thanks.

 

[Swapnil]

What you have said is exactly right. Let me express it in a more precise way. The Dirichlet's conditions are only sufficient, not necessary conditions. If a function f(t) meets these requirements then we know that we can express it as a Fourier series. However, if even if f(t) does not meet these requirements, we may still be able to express it as a Fourier series.

 

[AlephZero]

There are two possible meanings of "represent" in your post. The Weak Dirichlet condition (integral of the absolute value of the function is finite) says the Fourier series exists - i.e. you can calculate all the coefficients because the all the integrals that define them are finite.

The Strong Dirichlet condition (a finite number of extrema and a finite number of finite discontinuties) implies the Fourier series also converges to the original function (except at the discontinuities).

As Swapnil said these are not necessary conditions, and there are counterexamples. E.g the periodic function defined by x sin (1/x) in the interval -pi to pi has an infinite number of extrema, but there's no obvious reason (at least to me) why it doesn't have a convergent Fourier series.

 

[Benny]

I'm suprised that someone decided to answer this question after such a long time. I was referring to the "strong" Dirichlet conditions but the source from which my question arose didn't really specify the type of Dirichlet conditions (but the type was implied), which is why I didn't state the specific conditions. Anyway, I found the answer to my question a while ago but thanks for providing more extended answers.