Let $ f $ be a function on $ \mathbb{T}=[0,1] $ ($ 1 $-periodic) with bounded variation. Prove that if $ \widehat{f}(k)=\int_0^1f(x)e^{-2\pi ikx}dx=o(1/|k|) $, then $ f\in C(\mathbb{T}) $. I do not know how to use the assumption that $ f $ is of bounded variation. Can you give me some hints?
Asked
Active
Viewed 156 times
1
-
If $f$ has finite variance it means it has derivative (almost everywhere) and derivate is in $L^1$. Then integrate by parts and use the fact that Fourier coefficients of $L^1$ function tend to $0$. Maybe writing integral as a sum and using changing of order of summations would be simpler but this is the same idea. – Salcio Sep 05 '22 at 14:34
-
Yes, $f'\in L^1$. But that integration by parts is not valid unless $f$ is absolutely continuous; here the relevant distribution-wise derivative of $f$ is a measure... – David C. Ullrich Sep 05 '22 at 14:43
-
The last part of the following answer should be very helpful: https://math.stackexchange.com/a/467925/151552 – PhoemueX Sep 05 '22 at 22:35