Timeline for Hausdorff dimension of the graph of a BV function (in 1 dimensional setting)
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 20, 2019 at 14:50 | vote | accept | Riku | ||
| Apr 12, 2019 at 20:08 | comment | added | Anthony Quas | Because if the were more than M/r points with jump r, then the total variation would be more than M/r. * r | |
| Apr 12, 2019 at 9:33 | comment | added | Riku | I guess they are standard, but I've never seen the proofs. For (1), it should follow by considering that a BV function is sum of a decreasing and an increasing function. But what about (2)? | |
| Apr 12, 2019 at 1:28 | comment | added | Anthony Quas | The facts you’re asking for are standard. For 3, the image under projection onto the first coordinated has Hausdorff dimension 1, and projections don’t increase dimension (you can just project the covering to get a new covering) | |
| Apr 11, 2019 at 21:31 | comment | added | Riku | Thank you. 1) How do you prove that the function has at most countably many discontinuities, which are necessarily of jump type? 2) Why are there at most M/r points at which the function jumps by r or more? 3) You conclude that $\dim graph(f) \le 1$. How do you show that it is also $\ge 1$? | |
| Apr 11, 2019 at 21:16 | history | answered | Anthony Quas | CC BY-SA 4.0 |