Is there any $L_1$ function such that its fourier transform is in $L_1$ and the fourier transform is nowhere differentiable? Or every such Fourier transform must be differentiable a.e.?
2026-04-03 01:14:58.1775178898
Nowhere differentiable Fourier transform
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Let $f$ be a bounded, nowhere differentiable function on $\mathbb{R}$. Let $g$ be a standard normal pdf. Then $gf$ is nowhere differentiable and in $L^2$. Also $\hat{(gf)} = \hat{g} * \hat{f} $. The fourier transform of a Gaussian is Gaussian, and convolving with a smooth function makes a smooth function, so $\hat{(gf)}$ is smooth and in $L^2$. So this exhibits a smooth function whose Fourier transform is nowhere differentiable. I think this is correct (?). Does it answer your question?