Timeline for Reference request: Extensions of Wiener's Tauberian Theorem
Current License: CC BY-SA 4.0
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| when toggle format | what | by | license | comment | |
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| Jun 7, 2019 at 7:12 | history | edited | Kostya_I | CC BY-SA 4.0 |
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| Jun 7, 2019 at 6:33 | history | edited | Kostya_I | CC BY-SA 4.0 |
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| Jun 6, 2019 at 17:33 | comment | added | Kostya_I | But, perhaps, even more serious issue is that Bochner's theorem requires $\hat{g}/\hat{f}$ to be positive definite, which is a restrictive condition, and I see no reason why a ratio of two positive definite functions should be positive definite. | |
| Jun 6, 2019 at 17:29 | comment | added | Kostya_I | In fact it follows from the second part of the post that the step function example generalises to any non-negative $f$ and $g(x)=a^{-1}f(ax)$ with $0<a<1$. Indeed, in that case it is not hard to see that there will be a point where $|\hat{g}|>|\hat{f}|$, but a Fourier transform of probability measure is $\leq1$ by absolute value. | |
| Jun 6, 2019 at 14:52 | comment | added | JohnA | Can you give some more thoughts on why this is "very rare"? The example with the step function seems rather particular, and it is not clear to me that this generalizes. | |
| Jun 6, 2019 at 12:07 | history | answered | Kostya_I | CC BY-SA 4.0 |