Skip to main content
MathOverflow

Return to Question

Notice removed Draw attention by CommunityBot
Bounty Ended with no winning answer by CommunityBot
Notice added Draw attention by kaleidoscop
Bounty Started worth 50 reputation by kaleidoscop
miswrote the assumption
Source Link
kaleidoscop
kaleidoscop
  • 1.4k
  • 7
  • 16

This is another question on the possible shape of sets $A,B\subset \mathbb{R}^d,d\geq 2,$ where resp. a non-null Schwarz function $f$ and its Fourier transform can vanish.

A nice remark by Christian Remling is that choosing separable functions yields that both functions can vanish on a half-space. Can a function and its Fourier transform both vanish on a convex cone?? The theorem by Shapiro mentioned there says that $A$ and $B$ cannot be resp. a "major convex cone" (strictly larger than a half space) and a non-empty open set.

Hence a remaining question is what can happen if $A$ is a "minor cone", say the quadrant of points with non-negative coordinates. As we saw, $B$ can be a half-space, but not a major cone. Is it possible that $B$ has a "gap in each coordinate"?

Here is a one-sentence question: is it possible to find a non-null Schwartz function $f$ on $\mathbb R^2$ which vanishes on a quarter-space and which Fourier transform $\hat f(x,y)$ vanishes onfor $O\times O'$$x\in O$ or $y\in O'$ where $O,O'$ are non-empty open sets of $\mathbb R$.

This is another question on the possible shape of sets $A,B\subset \mathbb{R}^d,d\geq 2,$ where resp. a non-null Schwarz function $f$ and its Fourier transform can vanish.

A nice remark by Christian Remling is that choosing separable functions yields that both functions can vanish on a half-space. Can a function and its Fourier transform both vanish on a convex cone?? The theorem by Shapiro mentioned there says that $A$ and $B$ cannot be resp. a "major convex cone" (strictly larger than a half space) and a non-empty open set.

Hence a remaining question is what can happen if $A$ is a "minor cone", say the quadrant of points with non-negative coordinates. As we saw, $B$ can be a half-space, but not a major cone. Is it possible that $B$ has a "gap in each coordinate"?

Here is a one-sentence question: is it possible to find a non-null Schwartz function $f$ on $\mathbb R^2$ which vanishes on a quarter-space and which Fourier transform vanishes on $O\times O'$ where $O,O'$ are non-empty open sets of $\mathbb R$.

This is another question on the possible shape of sets $A,B\subset \mathbb{R}^d,d\geq 2,$ where resp. a non-null Schwarz function $f$ and its Fourier transform can vanish.

A nice remark by Christian Remling is that choosing separable functions yields that both functions can vanish on a half-space. Can a function and its Fourier transform both vanish on a convex cone?? The theorem by Shapiro mentioned there says that $A$ and $B$ cannot be resp. a "major convex cone" (strictly larger than a half space) and a non-empty open set.

Hence a remaining question is what can happen if $A$ is a "minor cone", say the quadrant of points with non-negative coordinates. As we saw, $B$ can be a half-space, but not a major cone. Is it possible that $B$ has a "gap in each coordinate"?

Here is a one-sentence question: is it possible to find a non-null Schwartz function $f$ on $\mathbb R^2$ which vanishes on a quarter-space and which Fourier transform $\hat f(x,y)$ vanishes for $x\in O$ or $y\in O'$ where $O,O'$ are non-empty open sets of $\mathbb R$.

Source Link
kaleidoscop
kaleidoscop
  • 1.4k
  • 7
  • 16

Possible gaps for a function and its Fourier transform

This is another question on the possible shape of sets $A,B\subset \mathbb{R}^d,d\geq 2,$ where resp. a non-null Schwarz function $f$ and its Fourier transform can vanish.

A nice remark by Christian Remling is that choosing separable functions yields that both functions can vanish on a half-space. Can a function and its Fourier transform both vanish on a convex cone?? The theorem by Shapiro mentioned there says that $A$ and $B$ cannot be resp. a "major convex cone" (strictly larger than a half space) and a non-empty open set.

Hence a remaining question is what can happen if $A$ is a "minor cone", say the quadrant of points with non-negative coordinates. As we saw, $B$ can be a half-space, but not a major cone. Is it possible that $B$ has a "gap in each coordinate"?

Here is a one-sentence question: is it possible to find a non-null Schwartz function $f$ on $\mathbb R^2$ which vanishes on a quarter-space and which Fourier transform vanishes on $O\times O'$ where $O,O'$ are non-empty open sets of $\mathbb R$.