User talk:Merge

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&mdash; merge 18:50, 29 December 2005 (UTC)

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I see you already found the math wikiproject, good. :) Thank you for your careful and well-summarized edits to several math articles. Oleg Alexandrov (talk) 23:15, 28 December 2005 (UTC)

Scalar
Hello. You are doing great work, but I'm not sure I like your edit on Scalar (mathematics). I'm afraid that your version starts on a too high level. We should try to make the article, and especially the first sentence, understandable to as many people as possible. Things like vector spaces over a general field do not help (when we start to learn about vector spaces, they are always over the reals, or perhaps C). Unfortunately, I'm not sure I'll find time soon to see what I can do about it, and anyway, it's not nice to butcher somebody's contribution. Please keep this in mind, and if you feel like it, see whether you can improve on your edit. Do ask me (or Oleg Alexandrov, actually, preferably Oleg ;) ) if I'm not being clear. Keep up the good work! -- Jitse Niesen (talk) 20:35, 19 February 2006 (UTC)


 * Hi Jitse. Many thanks for the comment.  I fully agree with your sentiment and suspect I erred on the side of rigour as a result of the original author's apparent dislike for intuitive introductory sentences.  In fact this article has been due for a major refactoring for a while, so I've just tried to do this and also rewritten the intro in what I hope is a more accessible form.  I'd be grateful for your thoughts when you have the chance to take a look!   &mdash; merge 16:00, 20 February 2006 (UTC)

Hölder and Cesàro techniques
I have been advised to bring the following, which was initiated on Talk:Grandi's series, to your attention:

Some information
Concerning Hölder and Cesàro, for details one is referred to


 * E.W. Hobson, The theory of functions of a real variable and the theory of Fourier's series, Volume II, 2nd edition, reprinted (Cambridge University Press, 1950), section 44.

Historically, Hölder's approach dates from 1882 (Math. Annalen, Vol. XX, (1882), p. 535) and Cesàro's from 1890 (Bulletin des Sciences Math., Vol. XIV (1890), p. 114). It was Knopp (Grenzwerte von Reihen bei der Annäherung an die Convergenzgrenze, Inaugural Dissertation, Berlin 1907) who first showed that a series which is summable $$\ (H,k)$$ (that is summable in accordance with Hölder's definition, of order $$\ k$$) is necessarily summable $$\ (C, k)$$ (that is summable in accordance with Cesàro's definition, of order $$\ k$$). The converse was established by Schnee (Math. Annalen, Vol. LXVII (1909), p. 110) and by Ford (Amer. Journ. of Math., Vol. XXXII (1910), p. 315).

You may take the above information over in your reworking of the present and other related articles.

--BF 02:01, 9 December 2006 (UTC)


 * Thanks for the info! In the last few weeks, I've learned some of this, but I didn't know the historical details.
 * I'm not sure if this article should talk about the higher-order methods, but they should definitely appear in the more dedicated articles.
 * One thing worries me: what exactly did Hölder propose in 1882? One or two of my sources claim that the 1890 Cesàro sum was the first systematic method of summation beyond the convergent one. It's implied that theorems by Abel and Frobenius that we now think of as relating different summation methods weren't posed as such at the time, and early efforts by Leibniz and Euler don't count because they weren't stated in precise terms. So it's conceivable to me that Hölder's work was along the same lines; on the other hand, it could be a scoop of Cesàro. Anyone know? Melchoir 02:22, 9 December 2006 (UTC)

You are welcome. I would recommend you to consult the book by Hobson cited above (Dover has reprinted this book in 1957).

Briefly, the Hölder and Cesàro techniques are in principle distinct; it was not until 1909 that one knew that a series is summable $$\ (H, k)$$ if and only if it is summable $$\ (C, k)$$ and not until 1913 that one knew that the two methods are equivalent.

For completeness, let $$a_1 + a_2 + \dots$$ be the series to be considered. Let $$s_n = a_1 + \dots + a_n, n=1,2,\dots$$. The Hölder partial sums are $$\{ h_n^{(k)}\} $$ where $$h_n^{(1)} = (s_1+\dots+s_n)/n, $$ $$h_n^{(k)} = (h_1^{(k-1)} + \dots + h_n^{(k-1)})/n, k=2,3,\dots.$$ The Cesàro partial sums are $$\{ C_n^{(k)} \}$$ where $$C_n^{(k)} = [k! (n-1)!/(n+k-1)!] s_n^{(k)}$$ in which $$s_n^{(0)} = s_n, $$ $$ s_n^{(1)} = s_1^{(0)} + \dots + s_n^{(0)}, $$ $$s_n^{(k)} = s_1^{(k-1)} + \dots + s_n^{(k-1)}, k=2,3,\dots .$$ The above series is summable $$\ (H,k)$$ if $$\lim_{n\to\infty} h_n^{(k)}$$ exists. It is summable $$\ (C, k)$$ if $$\lim_{n\to\infty} C_n^{(k)}$$ exists. It can be shown that when $$s = \lim_{n\to\infty} s_n$$ exists, both $$\lim_{n\to\infty} h_n^{(k)}$$ and $$\lim_{n\to\infty} C_n^{(k)}$$ indeed exist for all $$\ k$$ and are equal to $$\ s$$. Hobson refers to $$ \lim_{n\to\infty} s_n$$ as ordinary sum and to $$\lim_{n\to\infty} h_n^{(k)}$$ and $$\lim_{n\to\infty} C_n^{(k)}$$ as conventional sums, which satisfy the consistency condition, that they yield the same result as the ordinary sum when applied to convergent series.

The proof for the complete equivalence of the two methods of summation (or of summability of order $$\ k$$) is given by Schur (Math. Annalen, Vol. LXXIV (1913), p. 447) and by Hahn (Monatshefte f. Math. u. Physik, Vol. XXXIII (1923), p. 135). For details (including the proof of the equivalence of the two methods) see sections 55-57 of the above-cited book by Hobson.

I must admit that there is some confusion in textbooks concerning the two methods. For instance, Whittaker and Watson (4th edition, reprinted, Cambridge University Press, 1962, Sec. 8.43) describe the Cesàro method in exactly the same way as Hobson describes the Hölder method; there is no reference to Hölder's method in the book by Whittaker and Watson. I believe that on this matter Hobson should be considered as the authority.

--BF 15:13, 9 December 2006 (UTC)


 * Huh. Actually, yes, I was under the apparently mistaken impression that the Cesàro methods were the iterated ones. By pure luck, it doesn't matter for this article, but if there's confusion to be cleared up then we should do it somewhere. You seem to have good references in front of you; could you be persuaded to start up Hölder summation and/or expand Cesàro summation with information on history and higher-order definitions? Melchoir 23:14, 9 December 2006 (UTC)

Thank you for your suggestion. Expansion of the article on the Cesàro summation technique is problematic, since one will have to ascribe what at present is ascribed to Cesàro to its rightful author Hölder. This, I believe, should be in the first place the responsibility of the person or persons who have initiated the article on the subject, not least for the fact that this or these persons should have the opportunity to defend their standpoint as expressed in their article. This is why I initiated the present Talk.

As regards starting up an article on the Hölder summation method, it seems that this requires, roughly speaking, only renaming the article Cesàro summation as Hölder summation. As the two articles are necessarily closely related, I prefer that the entire task be undertaken by the initiator(s) of Cesàro summation.

If you wish to modify the article on Cesàro summation and further to create a page on Hölder summation, please feel free to do so; in doing so, please also feel free and take over whatever you feel necessary from my earlier texts on this page. In such case, I shall check the two articles and possibly amend them if necessary. In the event that you decide to work on these two articles, please indicate the confusion that exists in the literature of the subject concerning the two summation methods (even despite the fact that they are mathematically equivalent).

--BF 16:12, 10 December 2006 (UTC)


 * Sorry, I don't think I'm going to work on those articles myself. FYI, Cesàro summation was written largely by User:Merge, who doesn't seem to be around much. This particular talk page (Talk:Grandi's series) is hard to find and seldom read except by me, so if you want to really raise the issue, I encourage you to leave a message at User talk:Merge, Talk:Cesàro summation, or Wikipedia talk:WikiProject Mathematics. Or you can just wait until I make this a Featured Article, at which point lots of people will visit this talk page, but there's no guarantee when that'll happen! Melchoir 18:24, 10 December 2006 (UTC)

Thank you for your recommendations.

--BF 19:21, 10 December 2006 (UTC)

I hope that you will consider to modify the article concerning Cesàro summation in the light of the above information.

--BF 19:39, 10 December 2006 (UTC)