Suppose you have a cut-off function $\varphi(x)$ such that $\widehat{\varphi}(\xi) \in C_0^\infty$ is even, real valued and non-negative with compact support on a symmetric interval $[-a,a]$ for $a>0$. Is it possible to define another cut-off function $\tilde{\varphi} = \tilde{\varphi}(2^\delta x)$, with $\delta >0$ such that $\tilde{\varphi}\varphi \approx \varphi^2$ for appropriately chosen $\delta$?
2026-04-02 08:18:03.1775117883
Cut-off functions and compactly supported Fourier transforms.
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