Shawsa7

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RBFs and completely monotone functions

Hi, I notice in your page about RBF interpolation that you wrote: "It has been shown that any function that is completely monotone will have this property, including the Gaussian, inverse quadratic, and inverse multiquadric functions."

I found this confusing because the first example (the Gaussian) is not completely monotone.

A bit of reading suggests that there is a change of variables involved in transforming the Gaussian into a completely monotone function—the Gaussian is positive definite and radial, while the transformed version is completely monotone. The texts I found online refer to sigma and r, where r = sqrt(sigma).

I'm not a mathematician so I'm on shaky ground here and don't think I should correct the page. Maybe you can verify what I found and make your own corrections, if indeed they're necessary? Thanks. Theoh (talk) 14:28, 8 August 2021 (UTC)

Hi Theoh! Good catch! You're absolutely right. The key property is strict positive definiteness as discussed in the citation (Fasshaur 2007). I've updated the page accordingly. I'm not very active on Wikipedia these days, and my research interests have shifted so I'm not actively working with RBFs at the moment. It sounds like you've done your research, so I would encourage you to make edits and contribute to the page. In particular, you could explain the connection to completely monotone functions now that there is no mention of it on the page. If there is anything you're uncertain about, feel free to start a discussion on the talk page or reach out to me again. Shawsa7 (talk) 01:15, 9 August 2021 (UTC)