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Apr
9
accepted Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code
Apr
6
comment Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code
I'm glad to know of this nice paper of Hinman. As for my question, I'll accept your answer of "folklore" in a few days if no references come up before then.
Apr
5
comment Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code
(I have a pdf now, thank you MO!)
Apr
5
comment Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code
Thank you for the reference. I couldn't find an electronic copy in the usual places, so my response will take a little longer. (If anyone has a pdf handy I would be grateful for the share.)
Apr
5
comment Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code
Liang Yu, thank you for the nice example. It is short and pleasing, but different from what I am looking for in two ways. First, the fact I wish to cite would give the exact degree of unsolvability for each $\alpha$, while this example leaves a few jumps' gap. (The gap can be closed by coding $0^{(\alpha)}$ into the $U$.) Second, I'm looking for a reference, not a proof sketch. When I need to cite folklore, I will generally write up a proof for the sake of the literature's completeness. However, I prefer to hope that someone wrote it already!
Apr
4
asked Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code
Jul
2
awarded  Curious
May
2
awarded  Nice Question
May
1
comment Is this property equivalent to Lusin's property (N) for continuous functions?
And what a marvelous book it is, both efficient and clear. I think I'll be buying a Dover copy myself. Thank you Christian!
May
1
accepted Is this property equivalent to Lusin's property (N) for continuous functions?
Apr
29
asked Is this property equivalent to Lusin's property (N) for continuous functions?
Feb
20
comment Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?
Saks does seem to have this, though it is not set aside as a theorem, at a first look-through I think all the pieces are there. It has a very pleasing parallel analysis of the narrow and wide cases in Chapter VIII, sections 1,4 and 5. Thank you!
Feb
20
accepted Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?
Feb
20
comment Error of midpoint method for functions that are not twice-differentiable
I meant with the checkbox, but in any case I'm glad it helped.
Feb
19
awarded  Yearling
Feb
18
comment Error of midpoint method for functions that are not twice-differentiable
There are certainly some hypotheses one could add to get rid of the kind of bad behavior above. For a very weak example, I don't know how to make an $O(1/n)$ convergence with an absolutely continuity restriction. If you're satisfied with my answer, though, I hope you will accept it :-)
Feb
18
comment Algebraic characterization of real differentiation
For example, the polynomials are dense in $C^1[a,b]$ for the topology of uniform convergence (Stone-Weierstrass). But the derivative operator is not continuous with respect to this topology. This example does not contradict your answer, nor does it directly address the OP. But the thing I am confused about is under what circumstances would one be able to use your answer to identify the derivative.
Feb
17
comment Algebraic characterization of real differentiation
With respect to what topology should the rational functions be dense and the operator continuous?
Feb
17
comment Error of midpoint method for functions that are not twice-differentiable
Now I think I understand your previous comments -- you are just as interested in the "$o$" as the "$1/n$". I added a nasty function for which the error fails to be $o(1/n)$.
Feb
17
revised Error of midpoint method for functions that are not twice-differentiable
Added an example which illustrates that $O(1/n)$ is an exact bound.