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• CommentRowNumber1.
• CommentAuthorTobyBartels
• CommentTimeFeb 13th 2017

I wanted to understand Borel's Theorem better, so I wrote out a fairly explicit proof of the one-dimensional case.

1. Perhaps to a working analyst that argument looks natural, but to me it is extraordinary that anybody could come up with that!

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeFeb 14th 2017
• (edited Feb 14th 2017)

Possible typo: should “$H_n x = 0$” be “$\phi^{(k-i)}(H_n x)=0$”?

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 14th 2017

I have changed

  [[falling power]]


to

  [[falling factorial]]


which seems to be what was intended.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeFeb 14th 2017

I myself usually call it a “falling power”, and I believe this is the term used by Graham, Knuth, and Patashnik, so I added that terminology to falling factorial as well as a few redirects.

• CommentRowNumber6.
• CommentAuthorTobyBartels
• CommentTimeFeb 15th 2017

I'm glad that we do have a page on the falling power after all. (To my mind, it's not so much a falling factorial as a partial factorial. Factorials are always falling, after all.)

Mike, thanks for catching the typo.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeFeb 15th 2017

I wasn’t aware of this result before; it surprised me somewhat!

• CommentRowNumber8.
• CommentAuthorDavidRoberts
• CommentTimeFeb 15th 2017
• (edited Feb 15th 2017)

This is a special case of a theorem that says when the restriction map $C^\infty(M) \to C^\infty(K)$, for $M$ a manifold and appropriate closed subspaces $K\subset M$, is a submersion. For $M=\mathbb{R}^n$ one has the improved result that it is a split surjection of Fréchet spaces.

The theorem at hand considers $K=\{0\}\subset \mathbb{R}=M$, and by smooth functions on a point one has to consider Whitney jets, which reduce to a mere sequence of numbers, but in general need to satisfy the kind of error estimates that Taylor’s theorem gives.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeFeb 16th 2017

David, might you have a spare minute to add this kind of comment to the entry?

• CommentRowNumber10.
• CommentAuthorDavidRoberts
• CommentTimeFeb 16th 2017
• (edited Feb 16th 2017)

Sure. I might be writing a short note about a small extension I have made (from Euclidean space, which is in the literature, to Riemannian manifolds of bounded geometry) at some point, so I should gather the relevant references etc before then :-)

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeFeb 16th 2017

I would have called it a shifted factorial, since the difference between it and $k!$ is that it starts at $x$ rather than at $k$.

• CommentRowNumber12.
• CommentAuthorTobyBartels
• CommentTimeFeb 16th 2017

Mike, yes, that also makes sense, especially when $x$ is not a natural number.

Of course, there's also the case when $k$ is negative; then it's a rising power and a shifted reciprocal factorial. (In particular, $1/k! = 0^{\underline{-k}}$.)