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 $n\geq 1, x,y\in \mathbb{R}^n$.

Note that, by Bochner's theorem, $f = \widehat{\mu}$ for some Borel probability measure $\mu$ on $\mathbb{R}$.

**Question.** If $f \in L^{p}$ for some $p < \infty$, must we have $f(x)=O(|x|^{-c})$ for some $c>0$?

Edit: Formerly, I had the condition $p>2$ instead of $p < \infty$. Thanks to Christian Remling for pointing out that the condition $p>2$ adds nothing and the condition $p<\infty$ is needed.

Context: This is a natural extension of this question: https://math.stackexchange.com/questions/2296804/lp-implies-polynomial-decay

I posted this first to MSE a few months ago, got several up-votes, but nothing helpful: https://math.stackexchange.com/questions/2306071/decay-of-positive-definite-functions-in-lp

I am following the advice of Cross posts to Math SE regarding cross-posting.

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