what's the difference between "convergent" and "reconstruct-able"? I am reading this book:
http://www.abdn.ac.uk/~mth192/html/maths-music.html
There is a sentence on page 54:
"However, the question of convergence of the Fourier series is not the
same as the question of whether the function f(θ) can be reconstructed from
its Fourier coefficients an and bn."
I failed to understand this.
Why they are different questions?
(Maybe this is merely because my native language is not English, if so, I feel apologetic.)
 A: I believe you are talking about pointwise convergence. There are many modes of convergence, for example convergence in $L^2$, then we would have that $\|S_n f - f\|_{L^2} \to 0$ as $n \to \infty$ where $S_n f$ are the partial sums of the Fourier series of $f$ up to $n$. This is a quite easy result as the exponentials $e^{i n x}$ form a Schauder basis for $L^2$, then we can just use the inner product and use some syntax manipulation magic to obtain the result.
A much harder question however is about the pointwise convergence of the Fourier series, that is $S_n f(x) \to f(x)$ for (all?) $x$ as $n \to \infty$. This has been settled in 1966 by Carleson in Carleson's theorem. This only gives pointwise convergence almost everywhere for $L^2$ functions to the function itself. Of course, certain classes of functions even give uniform convergence to the original function.
So, your question amounts to the following. Given a function $f$, we can compute its Fourier coefficients and then write down the candidate for its Fourier series. Using different means we can prove the convergence of this series. However, proving the convergence does not imply that it actually converges to the original function! Just as you would have with Taylor series.
But, sure, we have stuff like Carleson's theorem that makes life good. :-).
A: Convergence of Fourier series means the convergence of the sequence of partial sums $\{s_m\}$ (notation is as in your book). There are cases in which the Fourier series does not converge to any limit, But the sequence of partial sums $\{s_m\}$ is cesaro summable $(C,n)$ for $n$ greater than some non negative integer $\alpha$. Thus when the Fourier series (partial sums) does not converge there still is a chance for the sequence of partial sums being Cesaro summable, there by enabling us to have a chance to reconstruct the function using the Cesaro sum. 
Here the question is not about the difference between 'convergence' and 'reconstructible'. In the book the author while refering to convergence, he is having in his mind, the Fourier series, but when he is refering to 'reconstructable', he is hoping that atleast, the sequence of partial sums is cesaro summable.
