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Dustin G. Mixon
Dustin G. Mixon
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Let $F$ denote the Fourier transform over some group. What is known about the following quantity?

$$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$

Here, $|x|$ denotes the pointwise absolute value of $x$. We know $\gamma\leq1$ since $x$ can be pointwise nonnegative. In the case where $F$ is the DFT, we also know $\gamma>0$ by a compactness argument, and computer simulations give $\gamma\leq0.72$ by taking $x$ to be some random bandlimited even function.

EDIT: Following Josep's comment, we actually know $\gamma\geq1/\sqrt{n}$ in the DFT case by passing to the 2-norm and applying Parseval.

I would mostly like to know if $\gamma\gg0$ (i.e., there exists a constant $\epsilon>0$ independent of $n$ such that $\gamma\geq\epsilon$), since this would imply that $\|Fx\|_1$ is small (i.e., $Fx$ is concentrated) only if $\|F|x|\|_1$ is also small. This would confirm my intuition that phase doesn't play much of a role in the concentration of a function's spectrum. Presumably, this question is natural enough to have been studied already.

Let $F$ denote the Fourier transform over some group. What is known about the following quantity?

$$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$

Here, $|x|$ denotes the pointwise absolute value of $x$. We know $\gamma\leq1$ since $x$ can be pointwise nonnegative. In the case where $F$ is the DFT, we also know $\gamma>0$ by a compactness argument, and computer simulations give $\gamma\leq0.72$ by taking $x$ to be some random bandlimited even function.

I would mostly like to know if $\gamma\gg0$, since this would imply that $\|Fx\|_1$ is small (i.e., $Fx$ is concentrated) only if $\|F|x|\|_1$ is also small. This would confirm my intuition that phase doesn't play much of a role in the concentration of a function's spectrum. Presumably, this question is natural enough to have been studied already.

Let $F$ denote the Fourier transform over some group. What is known about the following quantity?

$$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$

Here, $|x|$ denotes the pointwise absolute value of $x$. We know $\gamma\leq1$ since $x$ can be pointwise nonnegative. In the case where $F$ is the DFT, we also know $\gamma>0$ by a compactness argument, and computer simulations give $\gamma\leq0.72$ by taking $x$ to be some random bandlimited even function.

EDIT: Following Josep's comment, we actually know $\gamma\geq1/\sqrt{n}$ in the DFT case by passing to the 2-norm and applying Parseval.

I would mostly like to know if $\gamma\gg0$ (i.e., there exists a constant $\epsilon>0$ independent of $n$ such that $\gamma\geq\epsilon$), since this would imply that $\|Fx\|_1$ is small (i.e., $Fx$ is concentrated) only if $\|F|x|\|_1$ is also small. This would confirm my intuition that phase doesn't play much of a role in the concentration of a function's spectrum. Presumably, this question is natural enough to have been studied already.

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Dustin G. Mixon
Dustin G. Mixon
  • 7.6k
  • 2
  • 31
  • 56

Can phase significantly concentrate a function's spectrum?

Let $F$ denote the Fourier transform over some group. What is known about the following quantity?

$$\gamma:=\inf_{x\neq 0}\frac{\|Fx\|_1}{\|F|x|\|_1}$$

Here, $|x|$ denotes the pointwise absolute value of $x$. We know $\gamma\leq1$ since $x$ can be pointwise nonnegative. In the case where $F$ is the DFT, we also know $\gamma>0$ by a compactness argument, and computer simulations give $\gamma\leq0.72$ by taking $x$ to be some random bandlimited even function.

I would mostly like to know if $\gamma\gg0$, since this would imply that $\|Fx\|_1$ is small (i.e., $Fx$ is concentrated) only if $\|F|x|\|_1$ is also small. This would confirm my intuition that phase doesn't play much of a role in the concentration of a function's spectrum. Presumably, this question is natural enough to have been studied already.