Though we know for any $f\in L^1(R)$, its Fourier transform $\hat{f}\in C_0$, what can we say about its differentiability? I'm trying to construct an example to make both $f,\hat{f}\in L^1$ but $\hat{f}$ is nowhere differentiable. Is this possible?
Will the differentiability of Fourier transform be very bad?
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fourier-analysis
harmonic-analysis
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2The function $f_\delta(t)= \max(0, 1- |t|/\delta)$ is $L^1$ and has $\sup_\delta \Vert \widehat{f_\delta} \Vert_1 < \infty$. So I think you should be able to build an $f$ of the form you are seeking, by stacking together scaled and shifted copies of $\widehat{f_\delta}$. – 2017-02-11
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0@Yemon Choi Thank you! This is helpful. I will try to figure out this with your hint! – 2017-02-12
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0@Yemon Choi Could you please give more hint on the construction? I've been thinking on functions like $\sum2^n f_{3^n}(x-r_n)$, where $r_n$ is an enumeration of rational numbers, but I couldn't determine the differentiability of the function. – 2017-02-16