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It's all in the title :)

Theorem A' of Shapiro (see below) yields that if a tempered function $f$ on $\mathbb R^d$ has a spectral gap (i.e. $\hat f$ in the distributional sense vanishes on a non-empty open set) and $f$ vanishes on a "major" convex cone (for instance a cone of aperture $>\pi$ in $\mathbb R^2$) then $f=0$ identically.

He also mentions that the majority assumption is necessary as there exists a non-null function $f$ which vanishes on a half plane and has a spectral gap.

I really would like only to assume that $f$ vanishes on a minor cone (complement of a major cone), what need I assume on the spectral gap of $f$ to make sure $f$ is zero?

For instance: can both a tempered function $f$ and its Fourier transform vanish on a convex cone with non-empty interior?

H. A. Shapiro, Functions with a spectral gap, Bull. AMS, 79 (2), 1973

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    $\begingroup$ You can take $f(x,y)=g(x)h(y)$, with $g$ and $\widehat{h}$ both vanishing on a half line. $\endgroup$
    – Christian Remling
    Commented Nov 16 at 19:31
  • $\begingroup$ Right... With Shapiro's remark I thought it was an intricate problem but you simplified it :) Thanks $\endgroup$
    – kaleidoscop
    Commented Nov 17 at 18:05

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