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I am reading this book:

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.)

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The Fourier series may converge to a function different from the original function. The original function is "reconstructed" if the Fourier series converges to it. – Arturo Magidin Mar 25 '12 at 3:23
As a simple example, the function which is equal to $1$ at $\theta = 0$ and equal to $0$ otherwise cannot be reconstructed from its Fourier coefficients, which are all equal to zero (so the Fourier series converges to the zero function). – Qiaochu Yuan Mar 26 '12 at 3:34
@Arturo : I agree with your comment, but after reading the page 54 in the book, I don't think thats what the author intended to convey. – Rajesh Dachiraju Mar 26 '12 at 3:52
up vote 2 down vote accepted

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. :-).

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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.

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