Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous positive-definite function with $f(0)=1$. Positive-definiteness of $f$ means $$ \sum_{i=1}^{n}\sum_{j=1}^{n}f(x_i-x_j)y_i y_j \geq 0 $$ for all $n\geq 1, x,y\in \mathbb{R}^n$.
Question. If $f \in L^{p}$ for some $p>2$, must we have $f(x)=O(|x|^{-c})$ for some $c>0$?
Note that, by Bochner's theorem, $f = \widehat{\mu}$ for some probability measure $\mu$ on $\mathbb{R}$.
Context: This is a natural extension of this question: $L^p$ implies polynomial decay?
I am posting this over at MO, since it has stood several months here with no progress. https://mathoverflow.net/questions/278988/decay-of-positive-definite-function-in-lp