Timeline for RKHSs containing constant functions
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 4, 2013 at 18:27 | comment | added | Mark Meckes | Mike: Your guess is correct --- this is actually the conclusion I want. | |
| Jun 4, 2013 at 16:59 | comment | added | Mike Jury | There is another condition in terms of the geometry of the embedding $X\to H$ via $e:x\to k_x$ (assuming the normalization $k(x,x)\equiv 1$), though I'm not sure it's any easier to check: The space $H$ contains the constant functions if and only if the image of $X$ under $e$ lies in an affine hyperplane of $H$. (If $w\in H$ is constant on $X$, then the $k_x$'s lie in an affine hyperplane orthogonal to $w$, and conversely.) Though maybe this is the conclusion you want rather than a condition you want to check... | |
| Jun 4, 2013 at 15:22 | history | edited | Mark Meckes | CC BY-SA 3.0 |
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| Jun 4, 2013 at 15:17 | comment | added | Mark Meckes | Mike: You're right, I was sloppy about which basis I was thinking about. I'll also add in the specific kernel I'm interested in. | |
| Jun 4, 2013 at 15:14 | comment | added | Mike Jury | Your edits appeared after I posted my answer, but I don't think you want $\delta_{xy}$ in the condition you give, rather you subtract off the kernel that is identically $1$ on $X\times X$. Also, for most of the kernels I'm familiar with this condition is not so hard to check; do you have a particular kernel that you're looking at? | |
| Jun 4, 2013 at 15:09 | answer | added | Mike Jury | timeline score: 5 | |
| Jun 4, 2013 at 15:04 | history | edited | Mark Meckes | CC BY-SA 3.0 |
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| Jun 4, 2013 at 14:14 | history | asked | Mark Meckes | CC BY-SA 3.0 |