$\int_{\mathbb{R}} |x|^{\frac{1}{2}} |f(x)|^2 dx < \infty$ and $\int_{\mathbb{R}} |\xi|^{\frac{1}{2}} |\hat{f}(\xi)|^2 d\xi = \infty$

Question

Is there a $f$ such that

$\int_{\mathbb{R}} |x|^{\frac{1}{2}} |f(x)|^2 dx < \infty$ and $\int_{\mathbb{R}} |\xi|^{\frac{1}{2}} |\hat{f}(\xi)|^2 d\xi = \infty$, where $\hat{}$ denote the Fourier transform.

Answer 1

If $$ f(x)=|x|^{-1/8}\,K_{1/8}(x), $$ where $K_a$ is the modified Bessel function of the second kind, then $$ \hat f(\xi)=C\,(1+\xi^2)^{-3/8} $$ for some constant $C$. $K_a$ has exponential decay at $\infty$ and $|\xi|^{1/2}|\hat f(\xi)|^2\sim|\xi|^{-1}$ as $\xi\to\infty$.