Talk:Classical orthogonal polynomials

message: Hi, Sir anonymous. Thnak you for the edits ... Please obtain a regular login for Wikipedia so that we know how to reach you for discussions, and start getting some sense of who you are by your interests.linas 14:57, 13 Apr 2005 (UTC)
thread_title: Anonymous edits

message: I have added the theorem that each polynomial has all
its roots real (I consider this to be a really nice fact.)
I have put its proof, and that of the recurrence relation,
into a subordinate page, in accordance with
Wikipedia:WikiProject Mathematics/Proofs.  I think it
works out well.
I am grateful to Michael Hardy for his recent improvements
(I'm very new to all this.)  Unfortunately, I had to revert
a tiny change to Lemma 2.  Pn is not just orthogonal
to lower polynomials in the series, it is orthogonal to any
polynomial of lower degree.  The theorem about all real roots
requires that.
William Ackerman 05:13, 5 January 2006 (UTC)
thread_title:  Proofs moved. 

message: I just noticed that the recent page I created for proofs
was not made subordinate to the orthogonal polynomials
page, but is subordinate TO THE TOP LEVEL OF THE WHOLE
WIKIPEDIA!!!!!!
Can someone who is skillful at moving pages and cleaning
up newbies' messes please (a) accept my apology and (b) fix
this?
Thanks.
William Ackerman 05:12, 5 January 2006 (UTC)
replies: text: I deleted that page title /proofs.  You should find a "move" button on your screen that you can use for this sort of thing. Michael Hardy 22:35, 5 January 2006 (UTC)
thread_title:  OOPS!!!! 

message: The article looks more thorough ... but a few critiques:  Its a bit on the didactic side, especially in the intro. The first paragraph should start "An ortho poly is xxx.." rather than "An inner product is xxx...".  That is, the very first paragraph should assume the reader knows what an inner product is. Later in the article, you can clarify its definition.  E.g. the first paragraph might read "An ortho poly is a sequence of polys that are pair-wise orthogonal with regard to an inner product defined on the poly's. Ortho poly's are square integrable and form a hilbert space, although the spaces formed in this way are not identical." Or something like that. Also first paragraph should hint at some history: e.g "ortho polys were developed during the late 19th/early 20th centuries during a search for the solution of the xxx problem." (they were, to resolve some dispute, although I no longer remember what).
 
Also, note we have articles:  recursion relation  and polynomial sequence which may be linked. linas 03:28, 8 January 2006 (UTC)
thread_title:  Restructuring 

message: OK, I have made some changes in the early paragraphs
as per Linas's suggestions.  Feel free to edit,
improve, re-work, etc.  (This is Wikipedia, you don't
need me to tell you that. :-)
I put in the bare minimum about Hilbert spaces.  I think
anything further would distract from what the article is
really about.  I'm not happy with the detail about "functions
of finite norm", since we don't go into a discussion of the
norm.  But leaving out that qualifier would be incorrect -- the
Hilbert space comprises only the functions of finite norm.
And explaining it would probably be a distraction.  So I went
with the principle that, if the reader doesn't know what a
Hilbert space is, s/he will ignore that sentence with no
ill effect.  Otherwise, s/he will know what the norm means.
I put in a few words about historical background from Szego's
book.  It didn't sound like the "settling a dispute" that
Linas referred to.  It would be interesting if Linas's
recollection turns out to be different from this, especially
if some racy dispute at a gambling table (where Pascal
made his discoveries) is involved.
William Ackerman 22:14, 12 January 2006 (UTC)
replies: text: Heh. I didn't exactly have visions of Pascal using ortho poly's to operate a shell game of some sort, but the thought made me smile. I'll review the article presently; I have too many thing I'm involved in. linas 23:42, 12 January 2006 (UTC)
replies: text: Actually, my own fantasy was along the lines of a bar bet.  "Hey, everybody, this guy claims that all solutions to Legendre's equation are orthogonal. I say he should prove it or buy everyone a drink." William Ackerman 20:30, 13 January 2006 (UTC)
thread_title:  Improvements at the beginning. 

message: OK, I think I'm done.  :-)
The stuff that I just put in about associated Legendre polynomials
and spherical harmonics is excessive for this article, but I
wanted everything to be logically connected.  I (or someone else)
will very likely move it out to the Legendre and/or
Associated Legendre and/or spherical harmonics page, taking care
to keep everything logically harmonized.
William Ackerman 00:47, 30 January 2006 (UTC)
thread_title:  Associated Legendre functions 

message: what about mentioning, say, orthogonal polynomials in several noncommuting variables? also, perhaps the 2 volumes on orthogonal polynomials by Barry Simon could be added as references. Mct mht 02:09, 5 April 2006 (UTC)
replies: text: I would be in favor of rewriting the article to make it an overview of the THEORY of orthonormal polynomials. The other alternative would be to create a new article. For me orthonormal polynomials are part of a game, involving measures, Jacobi-operators and mappings between the 3 things. ElMaison (talk) 02:51, 1 April 2008 (UTC)
thread_title:  Generalizations? 

message: It appears that User:EdJohnston and User:PAR are improving the references to Abramowitz & Stegun, in not-entirely consistent ways.  I'd like you two experts to come up with a consistent way to present this.  Part of the reason I'd like to see this is that A&S is the bible for certain areas of math, and the various orthogonal polynomials are in the center of those areas.
Ed J. re-did the reference in the Bessel Functions page.  I then copied his changes to Orthogonal polynomials and associated Legendre polynomials, intending to make similar changes to all the others later.  Then PAR changed ortho poly to something else.  So we need to figure out what to do.
The pages that refer to A&S, and should have consistent notation, ISBN number, Gov't printing office info, etc, are as follows.  The relevant chapter is 22 (page 773) except as noted.
*Orthogonal polynomials (done)
*Legendre polynomials (done)
*Jacobi polynomials (done)
*Gegenbauer polynomials (done)
*Chebyshev polynomials
*Laguerre polynomials
*Hermite polynomials
*associated Legendre polynomials (chapter 8) (done)
*Bessel Functions (chapters 9 and 10) (done)
William Ackerman 21:48, 24 September 2006 (UTC)
replies: text: My Dover paperback from home, from a zillion years ago, says it (the paperback itself) was published in 1965, unedited from the original (whatever that was, GPO?) of 1964. Its cover is green. A much more recent Dover paperback in the library here at work says it is the 9th Dover printing, made from the 10th Governemnt Printing Office printing, which was dated 1972. Its cover is blue. It also has a list of errata that it claims have been fixed since the original. By comparing that errata list with what's on-line, we could find the pedigree of the latter. William Ackerman 15:39, 25 September 2006 (UTC)

text: I looked around some more. Go to http://www.math.sfu.ca/~cbm/aands/. They (Simon Fraser University Math department) give a lot of information about what is present, and their errata page mentions all the various printings. They say that what they have is the 10th printing. William Ackerman 15:55, 25 September 2006 (UTC)
replies: text: Could you give a specific link to the errata page?  What I'm driving at, is knowing whether the Dover text is as clean and recent as the one printed by the government.  Dover's own web site is no help at all. (It pictures the blue-cover version but gives no publication date, no revision info, no nothing). Somewhere there's a rumor that the blue paperback is actually a revised printing that is dated 1981. EdJohnston 20:46, 25 September 2006 (UTC)
replies: text: Try http://www.math.sfu.ca/~cbm/aands/intro.htm#002.  Both it and the newer (blue) Dover paperback list errata. A few spot checks indicate that the web page lists more errata than the blue book, though the difference that I found may be just a failure to show that the error was fixed.  That is, on page 774, formulas 22.2.6 and 22.2.7, the web version show asterisks next to them, indicating that this is a repaired error.  The blue book doesn't have the asterisks, but the formulas themselves are identical, as far as I can tell.  The errata notice on the web shows a printing history, going only up to a 9th printing, in 1970. William Ackerman 21:24, 25 September 2006 (UTC)

text: I'd rather not, but if we do, we should have the 1972 10th printing as THE reference, the only one included in the cite:book template. Reference to the earlier edition(s) should be as an aside, and it should be clear that these earlier editions are not what the article text is referencing, but rather are included for the readers convenience. PAR 03:26, 28 September 2006 (UTC)
replies: text: There are dozens of articles with A&S references. They should all be converted to a template, so that the layout, etc. could be centrally controlled. Much neater than editing dozens of articles to make the same change over & over. BTW, in my opinion, ordering info, etc. should not be given as part of the ref; the reader can be directed to article we have on A&S for that. linas 03:32, 4 October 2006 (UTC)
thread_title:  Abramowitz & Stegun references 

message: This page is listed on Wikipedia:WikiProject_Mathematics as needing a higher-level explanation.  I don't know how old that assessment is.  Do people still consider it a valid criticism? William Ackerman 22:17, 29 September 2006 (UTC)
replies: text: Well the orthogonal polynomials just fall out of the sky at the beginning of the article.  Perhaps there could be an example that's a physical problem?  EdJohnston 22:58, 29 September 2006 (UTC)

text: re "...needing a higher-level explanation...": yes, badly, IMHO. Mct mht 00:42, 30 September 2006 (UTC)
thread_title:  Needs higher-level explanation? 

message: La mise en place des tableaux pages 14-16 doit être revue car à l'impression la colonne de droite est tronquée. (VIDIANI Dijon 22 2 07)
The complaint is that the tables at the end are two wide to fit on this person's screen.  I already rearranged the tables some time ago to make them fit on my screen.  Does anyone have any ideas?
William Ackerman 18:01, 22 February 2007 (UTC)
thread_title:  Tables too wide? 

message: Firstly thanks to all the previous authors for their marvellous work in producing this useful page.  I can see the dilemna in how to present the classical theory (the one most commonly encountered) while acknowledging other formulations of the theory, in a manner that is accessible to the general reader.
What I've done is added some preamble to explain/motivate the definitions, and separated the classical definition from the generalizations.
* Since many of the proofs use the angle-bracket/inner product notation (even if only as a compact notation) it seems natural to begin with this.
* Next there are one-line descriptions of the classical definition and some generalizations to motivate the reader.
* A key point made, is that subtle changes in axioms/definitions can make big differences in the resulting properties of the polynomials, and that most of the properties discussed in the main article assume the classical definition.  Some specific examples not mentioned are the form of the recurrence relations and the distribution of roots.
* The classical definition is given its own subsection but otherwise unchanged.
* A new subsection on generalizations gives a brief glimpse into the various alternative theories, to motivate interested readers rather than attempting an exhaustive treatment which would take many pages to cover.
BPets (talk) 12:29, 19 September 2009 (UTC)
thread_title: Updated definition section

message: I agree with the comments in the "Generalizations?" thread above, that the alternative theories deserve more than a passing glance.  Also it is understandable that most readers primarily use the classical theory as "the" theory of orthogonal polynomials and would not want matters confused by mixing it with the generalizations/alternatives.  The article is currently very long and since the classical theory is a special case of the general theory, a large part of it should be moved to a separate article perhaps called "Classical orthogonal polynomials".  I certainly don't think any of this excellent work should be deleted.  Any thoughts on this? BPets (talk) 13:08, 19 September 2009 (UTC)
thread_title: Proposed major restructure

message: I remember the term "Orthogonal Function" in a Transport Phenomena class which I took in graduate school in 1996; but aside from the obvious case of sine and cosine and seeing how an integral of two functions over a certain interval can equal zero I don't see what they are used for. Does anyone have examples in science or engineering which utilize orthogonal functions and what benefit "Gramm - Schmidt orthogonalization" has in Linear Algebra? Thank You.JeepAssembler (talk) 21:56, 27 February 2010 (UTC)
thread_title:  What Are They Used For? 

message: The section details the claims of a recent paper which is not well cited.
This is not appropriate material for an encyclopaedia.
Either it should be rewritten or removed.  —Preceding unsigned comment added by 94.173.128.71 (talk) 14:22, 24 March 2010 (UTC)
replies: text: This seems to me so clearly true that I went ahead and removed it. You can find the text from the page history, or from the  of the page. Hanche (talk) 22:09, 24 March 2010 (UTC)
thread_title:  Riemann xi-function 

message: the section below should perhaps find its place in the new orthogonal polynomials article. Sasha (talk) 04:05, 25 August 2011 (UTC)
thread_title:  section from article 

message: What is this for a bad wording?
These pair of numbers seems to be the end points of the interval of their domain which is a (maybe trivial) subset of \R (respectifily its two-points compactification).
If you make your own problems here by extenting their domains always to \R (like for the Chebychev polynomials), then be told: limits are a total different thing/wording in mathematics! Sounds like a working physician has writen up this into the article. :-) But then he must define newly sub-intervals for their orthogonality here which only shows the nonsense to force definition on total \R for every orthogonal polynomal family. Regards. 78.94.123.114 (talk) 20:20, 23 September 2012 (UTC)
thread_title:  Limits of orthogonality