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Given a function ${f}(x)$ , which is continuous in the region $M_1<x<M_2$. Let ${F}(x)$ be the representation of ${f}(x)$ in Fourier series , such that

${f}(x)\approx{F}(x)$ = $a_0\over2$$+\sum_{n=1}^ma_n\cos(nxw)+\sum_{n=1}^mb_n\sin(nxw)$

Then is it possible to find the minimum value for $n$ such that $t_1$<${f}(x)$-${F}(x)$<$t_2$ where $t_1,t_2$ are real numbers. To be more specific, I am interested to know, how to find the value of $n$ where the condition

$-1$<${f}(x)$-${F}(x)$<$1$ will be satisfied for a given function ${f}(x)$ .

I don't know exactly whether the expression of my doubt is mathematically correct, So in sentential form, I like to know how fast (that is at what value of n) ${F}(x)$ reaches almost near to (or within bounds of ${(-1,1)}$) for given ${f}(x)$.

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When you say you are looking for a value for $n$, do you really mean $m$? Because $n$ is just the (bound) variable in the sum $\sum_{n=1}^m$ and is used nowhere else in your question, as far as I can seem. – Thomas Andrews Feb 19 '13 at 14:31
It seems to me you are asking the wrong question. Fourier series do not necessary converge pointwise. Rather, it is that the integral of the square of the difference between the function and the series over the interval vanishes as the number of terms in the series gets large. – Ron Gordon Feb 19 '13 at 14:39
@ThomasAndrews: You are right. Actually its the value of $m$ I was inquiring about. – smslce Feb 19 '13 at 15:00

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