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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 nx=0H_n x = 0” be “ϕ (ki)(H nx)=0\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 (M)C (K)C^\infty(M) \to C^\infty(K), for MM a manifold and appropriate closed subspaces KMK\subset M, is a submersion. For M= nM=\mathbb{R}^n one has the improved result that it is a split surjection of Fréchet spaces.

    The theorem at hand considers K={0}=MK=\{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!k! is that it starts at xx rather than at kk.

    • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 16th 2017

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

    Of course, there's also the case when kk is negative; then it's a rising power and a shifted reciprocal factorial. (In particular, 1/k!=0 k̲1/k! = 0^{\underline{-k}}.)

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