Timeline for Square-integrable unbounded function
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Mar 23, 2020 at 3:59 | vote | accept | Zurab Silagadze | ||
| Mar 22, 2020 at 16:38 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Mar 22, 2020 at 16:25 | comment | added | Iosif Pinelis | @ZurabSilagadze : (i) With the (standard, I think, and now explained) meaning of $\asymp$, positive real factors don't matter. (ii) As was stated, $c$ is any (constant) real number. (iii) I cannot use $h_k=1/k$ if $b\le2$, because I need $k^b h_k^2$ to be much greater than $\ln k$ for large $k$ -- in particular, to get the bound $O(1/k^c)$ for all real $c$. | |
| Mar 22, 2020 at 16:18 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Mar 22, 2020 at 16:09 | comment | added | Zurab Silagadze | @ Iosif Pinelis I think prefactor in your solution should be $(k\pi)^4$, not $k^4$. What is $c$ in the general case and why you cannot use $|x-k\pi|<1/k$ in this case also? | |
| Mar 22, 2020 at 15:30 | comment | added | Iosif Pinelis | @ZurabSilagadze : Perhaps, Richtmyer meant $a=1$ instead of $a=2$ (indeed, there is no reason to take $a=2$ to get an unbounded $f$, when $a=1$ would suffice). Then, still with $b=8$, we would have $2a-b/2=-2<-1$, just what is needed. | |
| Mar 22, 2020 at 15:23 | comment | added | Zurab Silagadze | Richtmyer says it is square-integrable, but it seems you are right (I have done some numerical calculations). The same statement is reproduced in arxiv.org/abs/quant-ph/9907069. Maybe there was a typo in Richtmyer. | |
| Mar 22, 2020 at 15:22 | comment | added | Pierre PC | Just another way of seeing it: instead of cutting the integral with respect to $h_k$, one can first change $\sin(x)^2$ to $Cx^2$ (or $\varepsilon x^2$, depending on the direction of the estimate), then perform the change of variable $h_k\cdot y=x-k\pi$. Then the integral between ± something of order $1/h_k$ is less than the integral over $\mathbb R$, and more than the integral between ± something of order 1. | |
| Mar 22, 2020 at 15:22 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Mar 22, 2020 at 15:11 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Mar 22, 2020 at 15:05 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Mar 22, 2020 at 14:44 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Mar 22, 2020 at 14:37 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Mar 22, 2020 at 14:14 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |