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From time to time or when reading a paper I hear the term "bootstrap" or "the bootstraping technique" or similar terminology. I cannot find a concise reference or explanation as to what is this method since when I google it I find all kinds of different things that are named as such. Is there a unifying philosophy that binds all these methods, techiniques, tricks, etc?

Sak
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  • Have you checked the wikipedia article? – Eff Jun 01 '17 at 23:07
  • Yes, thank you for the reference. The thing is that this is something concerning statistics and I have heard the term mainly in differential geometry, which is I feel unrelated. – Sak Jun 01 '17 at 23:09
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    Check this out https://math.stackexchange.com/questions/456140/what-is-elliptic-bootstrapping – Amitai Yuval Jun 01 '17 at 23:13
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    see http://www.phrases.org.uk/meanings/290800.html this is an old, old phrase... – Will Jagy Jun 02 '17 at 00:05
  • It makes sense that a technique which boosts the regularity of a function is called bootstraping with this meaning. Thank you that is very interesting. – Sak Jun 02 '17 at 01:22

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As is surely visible in many (older?) sources, the original colloquial sense of this was a partly ironic admonition to "lift yourself up by your own boot-straps" (bootstraps being loops or flaps at the sides of the top of boots that you'd pull on to help get your boots on). Of course, one cannot lift oneself by one's own bootstraps. I do not know how sarcastic or facetious this usage was. (I'm thinking of U.S. English, perhaps back to UK English.)

In various mathematical situations, what it really amounts to is induction, which can reasonably be portrayed as a semi-magical bootstrapping, after all, despite the fact that this is not possible "in the real world". For example, if $\Delta u = \lambda u$ with non-zero $\lambda$ and the Laplacian $\Delta$ on a manifold (e.g., the real line), and we know that $u$ is in some Sobolev space $H^s$, then since we also know that $\Delta$ maps $H^s$ to $H^{s-2}$ (for all $s$), solving such an equation maps $H^s$ to $H^{s+2}$. An induction shows that $u$ is in the nested intersection (projective limit) of the $H^s$'s, which, by Sobolev imbedding, consists of smooth functions. This is a way to prove a particular (important) instance of "elliptic regularity", for example.

paul garrett
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