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This is a stronger one related to the question Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x $.

$F_n(x) : [0,1] \rightarrow \bf R $, for $1 \leq i \leq n$, $F_n(x)= n\cdot g_{n,i}(x)$ if $x \in [\frac{i-1}{n}, \frac{i}{n})$, with $g_{n,i}$ a series of integrable functions. As $n, v \in \bf N$ goes to infinity simultaneously at the same rate, prove the convergence of
$\lim_{n,v \rightarrow \infty} \int_0^1 F_n(x) e^{-i2\pi v x}\,\mbox{d} x $ if $v/n$ is not an integer.

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    Clarification question: What is the reason for introducing $g_{n,i}$ with 2 indices? Why we cannot piece together $g$'s with the same $n$ and call it $g_n$?2012-06-09

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Let $F_n(x)=nx$. We can modify the values of $F_n$ at finitely many points so it satisfies the maximum and minimum conditions in the post. Then

$\int_0^1 F_n(x)e^{-i2\pi vx} dx=\int_0^1 nx e^{-i2\pi vx} dx=\frac{i}{2\pi}\frac{n}{v}$

and clearly the limit that you are interested in does not exist.

EDIT. One can also modify $F_n(x)$ at those finitely many points so that $F_n(x)$ is continuous for all $n$.

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    Than$k$s TCL, I cha$n$ged the condition.2011-01-09