Talk:Classical orthogonal polynomials

Please add new comments at the bottom of the page.

Long discussion
In case I make a mistake wo this, orthogonal polynomials (or any vectors in an inner product space) are orthogonal if  = &delta;ij (with the obvious inner product as defined in the article), correct or not? Dysprosia 10:08, 5 Sep 2004 (UTC)


 * No -- f and g are orthogonal if  = 0. In order that such a thing as Kronkecker's delta &delta;ij even be defined, the subscripts i and j have to mean something.  If one had a sequence of polynomials indexed by subscripts then one could say  = &delta;ij, and that would be the same as saying the members of that sequence are orthonormal (not orthogonal).  If you're talking about just two polynomials rather than about a sequence, then you have no "i and j.  Kronecker's delta is equal to 1 when i = j, and orthogonality of two polynomials has nothing to do with something being equal to 1. Michael Hardy 00:10, 6 Sep 2004 (UTC)

The first few lines say that two polynomials are orthogonal iff:

$$\int_{-\infty}^\infty f(x)g(x)w(x)\,dx=0. $$

this is not really true, this expression is simply one of many ways one can define a inner product space over polynomials. I beleive it wold be more correct to say two polynomials are orthogonal iff  = 0, where the inner product is defined arbitrarily. Nvrmnd 04:02, 26 Nov 2004 (UTC)

OK, that definition says two polynomials are always either proportional or orthogonal. Where does that get us? The definition is relative to a given w.

Charles Matthews 08:02, 26 Nov 2004 (UTC)

No, the definition  = 0 works for all inner product spaces but the current definition assumes that the inner product is always defined as $$\int_{-\infty}^\infty f(x)g(x)w(x)\,dx=0. $$ which is not true. Nvrmnd 18:06, 26 Nov 2004 (UTC)

No, you're in some way confused. Charles Matthews 18:18, 26 Nov 2004 (UTC)


 * Ok, allow me to clarify. I can define an inner product space between two polynomials an way I like providing that the 4 axioms are satisfied. Cleary, there is more than one way to define an inner product over continuous polynomials. As long as  = 0, f & g are orthogonal within this inner product space (this is mentioned in the article later). My argument is that the very first line "two polynomials f and g are orthogonal to each other with respect to a nonnegative "weight function" w precisely if (specific integral definition)" is misleading. Nvrmnd 18:44, 26 Nov 2004 (UTC)

The phrase

In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative "weight function" w precisely if ...

has to be parsed as

two polynomials f and g are orthogonal to each other with respect to a nonnegative "weight function" w

is the concept being defined, by the

... [formula with integral].

That is, it is a three way relation R(f,g,w) that gets defined; so if we write it


 *  = 0

that is really


 * w = 0

that we have defined as R. For obvious reasons the w is dropped from the notation.

There would be another way to say it, namely

given w, f and g are orthogonal to each other ...

and you might prefer this. At least, you might feel this is more accurate. But it's like currying on a variable - it makes no mathematical difference.

Hope that helps.

Charles Matthews 19:26, 26 Nov 2004 (UTC)

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)


 * To see why the definitions are important, the polynomials $$1,x,x^2,...$$ are orthogonal with respect to the inner product $$\langle x^i,x^j\rangle=\delta_{ij}$$ which cannot be expressed as a weighted integral over a real interval (otherwise $$0=\langle 1,x^2\rangle=\langle x,x\rangle=1$$). Many of the classical properties rely on the weighted integral definition however many other interesting properties pop up in other contexts.  See also the comments at the end of the page.  BPets (talk) 09:36, 19 September 2009 (UTC)

Limits on integrals of orthogonal polynomials
The limits are necessary, otherwise the integrals are indefinite integrals. exp(inx) for different n are orthogonal from negative infinity to infinity, but the indefinite integral of exp(i(n-m)x) is not zero.

Differential equation and Rodrigues formula
A cautionary remark: Both the second order differential equation (with coefficients g_1(x), g_2(x) independent(!) from n) and the Rodrigues type formula are only valid (in the form given here) for a very restricted class of weight functions and corresponding orthogonal polynomials including the classical series of Legendre, Jacobi or Hermite polynomials (given in the list of orthogonal polynomials in the article) and maybe a few others. In the case of the Rodrigues formula a necessary (but still not sufficient) constraint on the weight function w(x) is that the derivative w'(x) is of the form w(x)*R(x) where R(x) is a rational function. Therefore these paragraphs should be rewritten (or better omitted for the time being).


 * Hi anoymous, rather than removing text, why not add a few sentances at the bottom, along the lines of "more general ortho polynomials may be defined if blah,blah are allowed to blah,blah", mentioning that the 'axiom' of rodrigues formula must be modified/discard to define some general case. Let me make clear that a goal of this article is to accurately define the classical ortho poly case, so that its accessible to college students and "beginers"; then, additional info of interest to researchers can be attached to the bottom of the article, expounding on the general case.linas 20:07, 6 Apr 2005 (UTC)


 * Dear Linas, good suggestions. If it is the intention of the article to introduce here (mainly or exclusively) the "classical" orthogonal polynomials (which is a good idea for pedagocial and historical reasons) it should be stated more clearly in the intro of the article. In the present form the article first seems to deal with arbitrary weight functions which could be misleading if read in conjunction with the subsequent passages on three-term relation, differential equation and Rodrigues formula. The three-term relation is in fact valid for arbitrary weight functions but not the other two. So these properties could be better separated.


 * I personally do not know the theory of orthogonal polynomials to that level of detail; I would be unable to make this edit (without a good book in hand and some study). If you would be so kind ... Even considering just the classical poly's, this article is extremely brief, to the point of almost being useless.


 * The intention of the article is many-fold. Beginning student should be able to get a good overview; non-specialists, engineers & the like should be able to see a reasonably accurate reference, and mathematical researchers on the hunt for something should find a satisfying (and satisfyingly complete) presentation of the topic (although this might easily take multiple articles). One has to start somewhere.linas 01:25, 12 Apr 2005 (UTC)

Anonymous edits
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)

significant changes that I just made
This is my very first contribution to Wikipedia. I hope I haven't screwed up.

I have made some extensive edits, which see. I would like to do more. In particular, I would like to:

Present (and prove) the nice theorems that all polys have all their roots real, distinct, and in the interval of orthogonality, and that the roots of each one interlace the roots of the one below.

Present the actual Sturm-Liouville differential equations that give rise to the "classical" polys, and show how the terms appearing in the equations give rise to the 3 broad categories -- Hermite-like, Laguerre-like, and Jacobi-like.

Show the relationship between the terms in the differential equations and the weight functions and the items appearing in Rodrigues formula.

Discuss some of the applications -- Legendre for spherical harmonics, Hermite for quantum-mechanical harmonic oscillator (and other wonderful things), Laguerre for radial component of quantum mechanics of atoms, and Chebyshev for optimal polynomial approximations used in math libraries.

It is not my intention to write a "textbook", but I believe that the field of orthogonal polynomials needs a somewhat deeper treatment than other fields, because this field is something of an "orphan" in mathematics. It doesn't seem to be covered in math courses or textbooks, probably because it doesn't lead to important mathematical truths. It's just a fascinating sidelight. Placing some fairly comprehensive coverage here will fill a void in the (internet) literature.

I'm somewhat concerned about proofs of the theorems. The one that I put in (recurrence relation) is somewhat lengthy, though not lengthy by most math textbook standards. Also, the TeX notation that we have does not facilitate a really nice presentation. Some of the other proofs that I would like to put in will be somewhat more unwieldy. Perhaps putting them in separate pages, or an appendix, will help the flow of the main article. I solicit suggestions on this (and, obviously, on any other aspect of my plan.)

William Ackerman wba@alum.mit.edu William Ackerman 04:04, 30 December 2005 (UTC)


 * Welcome to Wikipedia! Your contributions are welcome! A couple of quick points, and suggestions, then:
 * Assume that 1 out of 10 readers have never heard of the concept, and that therefore the first 1-3 paragraphs should provide a synopsis of the entire concept (including a sentence or two about history) in a condensed but readable fashion. Assume these readers will stop reading after the 2nd paragraph.


 * Assume that 6 out of 10 readers have some of the basic concepts, but have forgotten, gotten rusty, or never had a good background. They're probably hunting for a useful formula, or some theorem. They probably will not read the entire article, start to end, but will read the parts in the middle that they need.


 * Assume that 3 out of 10 are more-or-less experts, but have misplaced a factor of 2 or some weight function or recurrence relation, and so are looking for a quick-reference guide.


 * You may also want to review the Manual of Style (mathematics). As to proofs, they are "experimental" at best right now, please review the contents of Category:Article proofs. I welcome further contributions. Direct further questions here, or to Wikipedia talk:WikiProject Mathematics linas 05:53, 30 December 2005 (UTC)

Proofs moved.
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 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)

OOPS!!!!
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)


 * 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)

Restructuring
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)

Improvements at the beginning.
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)


 * 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)


 * 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)

Associated Legendre functions
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)

Generalizations?
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)

this article gives a reader the misimpression that orthogonal polynomials are just some recursive formulas and solutions to some archane differential equations. even the simple fact that for every probability measure on the circle or on the line with finite moments, one can have a sequence of orthogonal polynomials is just hinted at, not spelled out. not a word about relationship to the moment problem, positive kernels, representation theory, etc. it needs some expert attention, and possilby a separate page with a more modern view. Mct mht 07:13, 4 May 2006 (UTC)


 * 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)

Rodriguez formula
I have reverted the link. Rodriguez formula simply redirects to orthogonal polynomials. The sad fact is, there is no Wikipedia page for Rodriguez formula!

Does anyone know more about this formula in its own right? Does anyone know who Rodriguez was? What the original publication was? What the actual statement of the formula is? Rodriguez formula is widely stated in books and papers on subjects to which it applies (for example, this WP page), but I've never seen any treatment of it in its own right. I don't even know whether it should properly be called "the Rodriguez formula", or "Rodriguez' formula", or whatever. Any ideas? William Ackerman 15:40, 12 September 2006 (UTC)


 * The spelling is "Rodrigues", after, I believe:
 * B. O. Rodrigues, "Mémoire sur l'attraction des sphéroïdes", Correspondance sur l’Ecole Royale Polytechnique T III, Paris, 1816, pp. 361–385.
 * Arfken and Weber (Mathematical Methods for Physicists) refer to it as the "Rodrigues representation" of the polynomial. I did a literature search, and many papers refer to it as a "Rodrigues formula". I haven't seen any examples of "Rodrigues' formula".  A couple of papers that might be worth looking up (but which unfortunately aren't available to me electronically), which present generalizations and reviews are:
 * Richard Rasala, "The Rodrigues formula and polynomial differential operators," Journal of Mathematical Analysis and Applications, v 84, n 2, Dec, 1981, p 443-482.
 * Saad Zagloul Rida and Ahmed M.A. El-Sayed, "Fractional calculus and generalized Rodrigues formula," Applied Mathematics and Computation, v 147, n 1, Jan 5, 2004, p 29-43.
 * —Steven G. Johnson 18:28, 12 September 2006 (UTC)

See also the entry 'Rodrigues Formula' in Springer's Encyclopaedia of Mathematics at EdJohnston 20:13, 12 September 2006 (UTC)
 * Shock! There is a WP article Olinde Rodrigues.  The article claims Rodrigues was Portuguese, contrary to its own first cited article  which is much more thorough and believes he was Spanish.  A lengthy discussion of Rodrigues' mathematical work is included in Simon Altmann's book, "Rotations, quaternions and double groups", which is the second reference of the Olinde Rodrigues article. EdJohnston 22:09, 12 September 2006 (UTC)


 * Fantastic! William Ackerman 23:07, 12 September 2006 (UTC)

Abramowitz & Stegun references
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. William Ackerman 21:48, 24 September 2006 (UTC)
 * 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)


 * 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)


 * 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)
 * 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)


 * 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)

Ok, before my edit, this was the situation:


 * 1) cite book: 1965 version (ISBN 0-486-61272-4)
 * 2) web link to 1972 version, chapter 22
 * 3) note about 1972 version
 * 4) link to ordering page of 1972 version

I wanted to remove all reference to the 1965 edition which is superceded by the 1972 version, so after my edit it was:


 * 1) cite book: 1972 version (ISBN 0-16-000202-8)
 * 2) web link to 1972 version, chapter 22
 * 3) link to ordering page of 1972 version

I'm still not clear on what the problem is. PAR 02:00, 26 September 2006 (UTC)

In a bookstore copy of the Dover edition, I found that it's got *still* a 1965 publication date, but it includes a copy of an Introduction dated 1970 to the 9th printing of the GPO edition, signed by Lewis Branscomb. Some articles within the Dover edition seem to have post-1965 references, contrary to the 1965 labelling. This leaves the hardback with a realistic date of 1972 and billed as the 10th printing, with a slight advantage on recency. So, I suggest that we supply both ISBNs since the paperback is so little different from the hardback, and it is so much cheaper. EdJohnston 17:19, 27 September 2006 (UTC)


 * 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)


 * 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)

Template created for making references to Abramowitz and Stegun
OK, I've made the template. (I hadn't know that mere civilians could make templates!) It is Template:Abramowitz_Stegun_ref, invoked with 2 arguments -- the chapter and the page, as in. I've edited Jacobi and Gegenbauer to use same. The experts (that's you, Ed) can modify the template, though I think leaving out the ordering info is the right thing. William Ackerman 22:17, 5 October 2006 (UTC)


 * This version is fine with me. There is no need to add further ordering information.
 * Here's an example of invoking the template (using same parameters as given by Wm A. above):


 * This expands to:
 * (See chapter 22).
 * (See chapter 22).


 * When you click on the highlighted chapter 22 then it causes your browser to open up the online copy of A&S and view the page number (773) that was passed as the second parameter. The template requires you to always specify a chapter number and page number, that's its only limitation.
 * EdJohnston 03:12, 6 October 2006 (UTC)

I've added another template, Template:Abramowitz_Stegun_ref2, for use where two chapter citations are needed. (I couldn't figure out how to make one template do both. As a software engineer, this disappoints me greatly :-)  It is invoked with 4 arguments -- the chapter and the page for the first reference, and then the chapter and page for the second, as in. I've used it in a few more pages. William Ackerman 16:08, 6 October 2006 (UTC)

Needs higher-level explanation?
This page is listed on 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)
 * 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)


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

Notation
Hello, Though this is simply a question of notation, I have only seen inner products written using the Dirac Notation, , is this simply a matter of physics vs. math notation for inner product spaces? Also, when writing or editing an entry, is there a way to code using Tex? Thanks, Kate —Preceding unsigned comment added by Kateryan (talk • contribs)

Hi - a few points:
 * To enter a math equation in TeX - enclose in math brackets - just edit a page with some equations on it and look at how its done. For example, edit this talk page, and look at how this equation is done.


 * $$\frac{d}{dx}\,e^x=e^x$$


 * Whenever editing a page, don't type carriage returns unless you are starting a new paragraph.


 * Whenever editing a page - remember that if the first character of a line is blank, the statement will be shown verbatim inside a box. Your edit above has made this mistake.


 * Check out How to edit a page for a tutorial

PAR 00:55, 4 November 2006 (UTC)


 * 


 * $$\langle f(x) \mid g(x) \rangle \, $$

Michael Hardy 20:01, 6 November 2006 (UTC)

Getting to the actual content of your question: I assume you are referring to the use of a simple comma separating the two vectors (polynomials, whatever) vs. using a vertical stroke. While I'm not an expert in the details of these conventions, I believe it is in fact a math vs. physics thing. The vertical bar is the "bra-ket notation" used in quantum mechanics, which has its own elegant logic. That logic is, unfortunately, not consistent with the way mathematicians define the inner product with a complex field. See the inner product page, and particularly the remark at the bottom of the "definitions" section. There is a difference in which of the vectors is complex conjugated. So the use of comma vs. vertical bar is not just a matter of typographical preference. Now it only makes a difference if the underlying field is complex&mdash;since the orthogonal polynomials being described here are real, taking a complex conjugate of one or the other makes no difference. I think we should stick with the mathematicians' convention for this page. The pointer to the inner product page leads to bra-ket notation in any case.

I learn something new and interesting every time I come to Wikipedia! William Ackerman 22:34, 6 November 2006 (UTC)

Tables too wide?
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)

Updated definition section
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. BPets (talk) 12:29, 19 September 2009 (UTC)
 * 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.

Proposed major restructure
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)

Error in ortho polys for measures
I believe there is a missing normalization of

$$ \frac1{\sqrt{D_n D_{n-1}}}$$

where $$D_n$$ is the Hankel determinant of order n given by the determinant of the matrix of moments $$(m_{(i+j)})_{0\le i,j \le n}$$, where $$m_k = \int x^k \mu(dx)$$. See http://www.williams.edu/go/math/sjmiller/public_html/book/papers/jcmp.pdf.

What Are They Used For?
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)JeepAssemblerJeepAssembler (talk) 21:56, 27 February 2010 (UTC)

Riemann xi-function
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)
 * 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)

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

General properties of orthogonal polynomial sequences
All orthogonal polynomial sequences have a number of elegant and fascinating properties. Before proceeding with them:

Lemma 1: Given an orthogonal polynomial sequence $$p_i(x)$$, any nth-degree polynomial S(x) can be expanded in terms of $$p_0, \dots, p_n$$. That is, there are coefficients $$\alpha_0, \dots, \alpha_n$$ such that


 * $$S(x)=\sum_{i=0}^n \alpha_i p_i(x).$$

Proof by mathematical induction. Choose $$\alpha_n$$ so that the $$x^n$$ term of S(x) matches that of $$\alpha_n P_n(x)$$. Then $$S(x)-\alpha_n P_n(x)$$ is an (n &minus; 1)th-degree polynomial. Continue downward.

The coefficients $$\alpha_i$$ can be calculated directly using orthogonality. First multiply $$S$$ by $$p_k$$ and weight function $$W$$, then integrate:


 * $$\int S(x) p_k(x) W(x) \, dx = \sum_{i=0}^n {\alpha}_i \int p_i(x) p_k(x) W(x) \, dx = \alpha_k \int p_k^2(x) W(x) \, dx,$$

giving


 * $$\alpha_k = {\displaystyle\int S(x) p_k(x) W(x) \, dx \over\displaystyle\int p_k^2(x) W(x) \, dx}. $$

Lemma 2: Given an orthogonal polynomial sequence, each of its polynomials is orthogonal to any polynomial of strictly lower degree.

Proof: Given n, any polynomial of degree n &minus; 1 or lower can be expanded in terms of $$p_0, \dots, p_{n-1}$$. Polynomial $$p_n\,$$ is orthogonal to each of them.

Minimal norm
Each polynomial in an orthogonal sequence has minimal norm among all polynomials with the same degree and leading coefficient.

Given n and any polynomial p(x) of degree n with the same leading coefficient can be expanded as


 * $$p(x) = p_n(x) + \sum_{i=0}^{n-1} \alpha_i\ p_i(x).$$

Using orthogonality, the squared norm of p(x) satisfies


 * $$\|p(x)\|^2 = \langle p(x),p(x)\rangle = \|p_n(x)\|^2 + \sum_{i=0}^{n-1} \alpha_i^2\|p_i(x)\|^2 \ge \|p_n(x)\|^2. $$

Since the norms are positive, take the square roots of both sides and the result follows. An interpretation of this result is that orthogonal polynomials are minimal in a generalized least squares sense. For example, the classical orthogonal polynomials have a minimal weighted mean square value.

Recurrence relations
Every orthogonal sequence has a recurrence formula relating any three consecutive polynomials in the sequence:


 * $$p_{n+1} = (a_nx+b_n) p_n - c_n p_{n-1}. \, $$

The coefficients a, b, and c depend on n, as well  as the standardization. Favard's theorem shows conversely that a sequence of polynomials satisfying such a recurrence relation is an orthogonal sequence.

We will prove this for fixed n, and omit the subscripts on a, b, and c.

First, choose a so that the $$x^{n+1}$$ terms match, so we have


 * $$a x p_n - p_{n+1} = \text{a polynomial of degree }n. \, $$

Next, choose b so that the $$x^n$$ terms match, so we have


 * $$(ax+b) p_n - p_{n+1} = \text{a polynomial of degree }n - 1 \, $$

Expand the right-hand-side in terms of polynomials in the sequence


 * $$(ax+b) p_n - p_{n+1} = \sum_{i=0}^{n-1} \lambda_i p_i$$

Now if $$j\le n-1$$, then


 * $$\langle (ax+b) p_n, p_j \rangle - \langle p_{n+1}, p_j \rangle = \sum_{i=0}^{n-1} \lambda_i \langle p_i, p_j \rangle = \lambda_j \langle p_j, p_j \rangle.$$

But


 * $$\langle p_n, p_j \rangle = 0\text{ and }\langle p_{n+1}, p_j \rangle = 0, \, $$

so


 * $$a \langle x p_n, p_j \rangle = \lambda_j \langle p_j, p_j \rangle. \, $$

Since the inner product is just an integral involving the product:


 * $$\langle x p_n, p_j \rangle = \langle p_n, x p_j \rangle \, $$

we have


 * $$a \langle p_n, x p_j \rangle = \lambda_j \langle p_j, p_j \rangle \, $$

If $$\ j < n-1$$, then $$x\ p_j$$ has degree $$\le n-1$$, so it is orthogonal to $$\ p_n$$; hence $${\lambda}_j \langle p_j,\ p_j \rangle\ =\ 0$$, which implies $${\lambda}_j\ =\ 0$$ for $$\ j < n-1$$.

Therefore, only $$\lambda_{n-1}$$ can be nonzero, so


 * $$(ax+b) p_n - p_{n+1}\ = \lambda_{n-1} p_{n-1} \ $$

Letting $$c = \lambda_{n-1}$$, we have


 * $$p_{n+1} = (ax+b) p_n - c p_{n-1}. \, $$

The values of $$a_n$$, $$b_n$$ and $$c_n$$ can be worked out directly. Let $$k_j$$ and $$k_j'$$ be the first and second coefficients of $$p_j$$:


 * $$p_j(x)=k_jx^j+k_j'x^{j-1}+\cdots \, $$

and $$h_j$$ be the inner product of $$p_j$$ with itself:


 * $$h_j\ =\ \langle p_j,\ p_j \rangle.$$

We have


 * $$a_n=\frac{k_{n+1}}{k_n},\qquad b_n=a_n \left(\frac{k_{n+1}'}{k_{n+1}} -

\frac{k_n'}{k_n} \right), \qquad c_n=a_n \left(\frac{k_{n-1}h_n}{k_n h_{n-1}} \right). $$

If the polynomials are chosen to be normalized, $$h_j=1$$, then the recursion takes the symmetric form of the eigenvalue equation of an associated Jacobi operator


 * $$ \alpha_n p_{n+1} + \beta_n p_n + \alpha_{n-1} p_{n-1} = x\, p_n, \qquad \alpha_n = \frac{1}{a_n}, \, \beta_n = -\frac{b_n}{a_n}.$$

Existence of real roots
Each polynomial in an orthogonal sequence has all n of its roots real, distinct, and strictly inside the interval of orthogonality.

This follows from the proof of interlacing of roots below. Here is a direct proof.

Let m be the number of places where the sign of Pn changes inside the interval of orthogonality, and let $$x_1 \dots x_m$$ be those points. Each of those points is a root of Pn. By the fundamental theorem of algebra, m ≤ n. Now m might be strictly less than n if some roots of Pn are complex, or not inside the interval of orthogonality, or not distinct. We will show that m = n.

Let $$S(x) = \prod_{j=1}^m (x - x_j).$$

This is an mth-degree polynomial that changes sign at each of the xj, the same way that Pn(x) does. S(x)Pn(x) is therefore strictly positive, or strictly negative, everywhere except at the xj. S(x)Pn(x)W(x) is also strictly positive or strictly negative except at the xj and possibly the end points.

Therefore, $$\langle S, P_n \rangle$$, the integral of this, is nonzero. But, by Lemma 2, Pn is orthogonal to any polynomial of lower degree, so the degree of S must be n.

Interlacing of roots
The roots of each polynomial lie strictly between the roots of the next higher polynomial in the sequence.

First, standardize all of the polynomials so that their leading coefficients are positive. This will not affect the roots.

We use induction on n. Let n≥1 and let $$x_1 < \cdots < x_n$$ be the roots of $$P_n$$. Assuming by induction that the roots of $$P_{n-1}$$ lie strictly between the $$x_j$$, we find that the signs of $$P_{n-1}(x_j)$$ alternate with j. Moreover $$P_{n-1}(x_n)>0$$, since the leading coeffient is positive and $$P_{n-1}$$ has no greater zero. To summarize, $$(-1)^{n-j}P_{n-1}(x_j) > 0$$.

By the recurrence formula


 * $$P_{n+1}(x) = (ax + b) P_n(x) - c P_{n-1}(x)\,$$

with $$c = a\,\frac{k_{n-1}h_n}{k_{n}h_{n-1}} > 0\,$$ we conclude that $$(-1)^{n-j}P_{n+1}(x_j) = (-1)^{n-j+1}cP_{n-1}(x_j) < 0$$. By the intermediate value theorem, $$P_{n+1}$$ has at least one zero between $$x_j$$ and $$x_{j+1}$$ for $$j=1,\ldots,n-1$$. Additionally, $$P_{n+1}(x_n) < 0$$ and the leading coefficient is positive, so $$P_{n+1}$$ has an additional zero greater than $$x_n$$. For a similar reason it has a zero less than $$x_1$$, and the induction step is complete.

Constructing orthogonal polynomials by using moments
Let


 * $$ \mu_n = \int_\mathbb{R} x^n\,d\mu $$

be the moments of a measure &mu;. Then the polynomial sequence defined by


 * $$ p_n(x) = \det\left[

\begin{matrix} \mu_0 & \mu_1 & \mu_2 & \cdots & \mu_n \\ \mu_1 & \mu_2 & \mu_3 & \cdots & \mu_{n+1} \\ \mu_2 & \mu_3 & \mu_4 & \cdots & \mu_{n+2} \\ \vdots & \vdots & \vdots &     & \vdots \\ \mu_{n-1} & \mu_n & \mu_{n+1} & \cdots & \mu_{2n-1} \\ 1 & x & x^2 & \cdots & x^n \end{matrix} \right] $$

is a sequence of orthogonal polynomials with respect to the measure &mu;. To see this, consider the inner product of pn(x) with xk for any k < n. We will see that the value of this inner product is zero.



\begin{align} \int_\mathbb{R} x^k p_n(x)\,d\mu & {} = \int_\mathbb{R} x^k \det\left[ \begin{matrix} \mu_0 & \mu_1 & \mu_2 & \cdots & \mu_n \\ \mu_1 & \mu_2 & \mu_3 & \cdots & \mu_{n+1} \\ \mu_2 & \mu_3 & \mu_4 & \cdots & \mu_{n+2} \\ \vdots & \vdots & \vdots &     & \vdots \\ \mu_{n-1} & \mu_n & \mu_{n+1} & \cdots & \mu_{2n-1} \\ 1 & x & x^2 & \cdots & x^n \end{matrix} \right] \,d\mu \\ \\ & {} = \int_\mathbb{R} \det\left[ \begin{matrix} \mu_0 & \mu_1 & \mu_2 & \cdots & \mu_n \\ \mu_1 & \mu_2 & \mu_3 & \cdots & \mu_{n+1} \\ \mu_2 & \mu_3 & \mu_4 & \cdots & \mu_{n+2} \\ \vdots & \vdots & \vdots &     & \vdots \\ \mu_{n-1} & \mu_n & \mu_{n+1} & \cdots & \mu_{2n-1} \\ x^k & x^{k+1} & x^{k+2} & \cdots & x^{k+n} \end{matrix} \right] \,d\mu \\ \\ & {} = \det\left[ \begin{matrix} \mu_0 & \mu_1 & \mu_2 & \cdots & \mu_n \\ \mu_1 & \mu_2 & \mu_3 & \cdots & \mu_{n+1} \\ \mu_2 & \mu_3 & \mu_4 & \cdots & \mu_{n+2} \\ \vdots & \vdots & \vdots &     & \vdots \\ \mu_{n-1} & \mu_n & \mu_{n+1} & \cdots & \mu_{2n-1} \\ \displaystyle \int_\mathbb{R} x^k \, d\mu & \displaystyle \int_\mathbb{R} x^{k+1} \, d\mu & \displaystyle \int_\mathbb{R} x^{k+2} \, d\mu & \cdots & \displaystyle \int_\mathbb{R} x^{k+n} \, d\mu \end{matrix} \right] \\ \\ & {} = \det \left[ \begin{matrix} \mu_0 & \mu_1 & \mu_2 & \cdots & \mu_n \\ \mu_1 & \mu_2 & \mu_3 & \cdots & \mu_{n+1} \\ \mu_2 & \mu_3 & \mu_4 & \cdots & \mu_{n+2} \\ \vdots & \vdots & \vdots &     & \vdots \\ \mu_{n-1} & \mu_n & \mu_{n+1} & \cdots & \mu_{2n-1} \\ \mu_k & \mu_{k+1} & \mu_{k+2} & \cdots & \mu_{k+n} \end{matrix} \right] \\ \\ & {} = 0\text{ if } k < n,\text{ since the matrix has two identical rows}. \end{align} $$

(The entry-by-entry integration merely says the integral of a linear combination of functions is the same linear combination of the separate integrals. It is a linear combination because only one row contains non-constant entries. There is no problem with convergence because the sum is finite.)

Thus pn(x) is orthogonal to xk for all k < n. That means this is a sequence of orthogonal polynomials for the measure μ.

Limits of orthogonality
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)