Timeline for Build an explicit "small perturbation" of the identity satisfying some properties
Current License: CC BY-SA 4.0
9 events
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| Dec 10, 2021 at 0:39 | vote | accept | CommunityBot | ||
| Dec 9, 2021 at 21:40 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Dec 9, 2021 at 21:37 | comment | added | Iosif Pinelis | @Dal : Your original question has been fully answered, and I even answered all your extra questions. Also, multiple questions for one post are discouraged on MathOverflow, anyway. So, let us have a closure here. | |
| Dec 9, 2021 at 21:37 | comment | added | Iosif Pinelis | @Dal : No to (3) and (4) either. Indeed, if for $F:=f_\epsilon$ we have $F''\le CF'$ and $F'>0$ on an interval $(a,b]$, then on $(a,b]$ we have $(\ln F')'\le C$ and hence $\ln F'(x)-\ln F'(a)\le C(x-a)$ and $F'(x)\le F'(a)e^ {C(x-a)}$ for $x\in[a,b]$, which implies $F'\ge0$ on $[a,b]$ if $F'(a)=0$. So, we get a contradiction with the condition $F'>0$ on an interval $(a,b]$. Now take $a:=\max\{z\colon F(z)=\bar x\}$. | |
| Dec 9, 2021 at 21:19 | comment | added | Iosif Pinelis | @Dal : (1) I have added the explicit expression for $h$. (2) No, because a convex function which equals a constant $c$ on a nonzero-length interval must be $\ge c$ everywhere and therefore cannot approximate the identity function. | |
| Dec 8, 2021 at 10:35 | comment | added | user139844 | Other two questions: (3) what is the dependence of $f'_\epsilon$ on $\epsilon$? Is it true that $f''_\epsilon \le C_\epsilon f'_\epsilon$? (4) Also, can the construction be generalized to get a $C^3$ function? If yes, on this $C^3$ function, do we have a better control on the second derivative, e.g. something like $f''_\epsilon \le C_\epsilon f'_\epsilon$? | |
| Dec 8, 2021 at 7:38 | comment | added | user139844 | Thank you so much. I have a couple of questions: (1) how can one compute $h$ explicitly? (2) Is it possible for the function $f_\epsilon$ to be built also to be convex? | |
| Dec 7, 2021 at 22:54 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Dec 7, 2021 at 22:42 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |