Suppose $f\in C_0^{\infty}(\mathbb{R}^n ),$ then clearly we have supp$(f\ast f)\subseteq$ supp$(f)+$ supp$(f)$. The question is whether supp$(f\ast f)\subseteq 2$ supp$(f)$ holds? Any counterexample?
2026-03-06 03:37:27.1772768247
Support of Auto-correlation
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Counterexample $f=\chi_{[-2,-1]}+\chi_{[1,2]}$ then $0\in$ supp$(f\ast f).$ replacing $f$ with a proper smooth function we can fund a smooth compactly supported function that has $0$ in its support.