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 Jan29 awarded Supporter Jan9 comment Convergence of $\lim \limits_{n,v \rightarrow \infty} \int_0^1 F_n (x) e^{-i2\pi v x} \mbox{d} x$ Thanks TCL, I changed the condition. Jan9 revised Convergence of $\lim \limits_{n,v \rightarrow \infty} \int_0^1 F_n (x) e^{-i2\pi v x} \mbox{d} x$ deleted 34 characters in body Jan4 revised Convergence of $\lim \limits_{n,v \rightarrow \infty} \int_0^1 F_n (x) e^{-i2\pi v x} \mbox{d} x$ added 17 characters in body Jan4 comment Convergence of $\lim \limits_{n,v \rightarrow \infty} \int_0^1 F_n (x) e^{-i2\pi v x} \mbox{d} x$ If $n$ and $v$ goes to infinity at the same rate (which I should include in the conditions), that limit exists. Jan2 comment Convergence of $\lim \limits_{n,v \rightarrow \infty} \int_0^1 F_n (x) e^{-i2\pi v x} \mbox{d} x$ @TCL, That simple case has been answered by Nick Kirby in the question [Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x$] linked above. Jan1 awarded Student Jan1 asked Convergence of $\lim \limits_{n,v \rightarrow \infty} \int_0^1 F_n (x) e^{-i2\pi v x} \mbox{d} x$ Jan1 awarded Scholar Jan1 accepted Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x$ Dec31 comment Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x$ Yes I see your point, just mis-understood you. Dec31 comment Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x$ My feeling is, if $v/n$ is not integer, integration of the product $f_n(x)e^{-i2\pi v x}$ cannot be broken into a sum of $n$ integrals. Can you give more hints? Dec31 comment Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x$ It is the convergence in the two variables simultaneously. Dec31 revised Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x$ added 49 characters in body Dec31 revised Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x$ added 12 characters in body; added 26 characters in body; edited title Dec30 awarded Editor Dec30 revised Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x$ added 1 characters in body; edited title Dec30 revised Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x$ deleted 1 characters in body; added 3 characters in body Dec30 asked Convergence of $\lim_{n,v \rightarrow \infty} \int_0^1 f_n (x) e^{-i2\pi v x} \mbox{d} x$