View source for Talk:Chebyshev polynomials

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== Notes on transcription and correct reading==

Name of this guy in original ia Пафнутий Львович Чебышёв
Львович means that his father's name was Lev (There is a big cat from genus Panthera called lion)
Чебышёв - should be transcripted as Chebyshov
Ч should be read softly 
ё - here we have two dots over e so there cannot be e but o 
(Russians usually don't write these two dots but it does not mean that they are not there) 
Accent should be placed on the last syllable
If you dont believe me consult it with Russian speaking guy

==[Untitled]==
Michael, it's not actually your browser. You can set that in your wikipedia preferences - by default it will render simple equations in normal text. And you apparently just complicated the equation by adding some white space ;-) [[User:Snoyes|snoyes]] 23:37 Mar 1, 2003 (UTC)

You're right; it works.  Thanks.  Why is the default the option that renders things as normal text, given that that is never what is intended when things are set with TeX? [[User:Michael Hardy|Michael Hardy]] 23:45 Mar 1, 2003 (UTC)

:I don't know why that is the default option - best to ask one of the developers, maybe on the village pump. --[[User:Snoyes|snoyes]] 23:53 Mar 1, 2003 (UTC)

And BTW, there don't seem to be any other articles about special polynomial sequences on Wikipedia. [[User:Michael Hardy|Michael Hardy]] 23:45 Mar 1, 2003 (UTC)

----

//Here's the polynomial sequence on the page. [[User:Cyp|&#922;&#963;&#965;&#960; ''Cyp'']] [[User:Cyp/Will this eventually appear on Special:Wantedpages?| ]] [[User talk:Cyp|&nbsp;]] 19:47, 25 Jan 2004 (UTC)
<pre>#include <stdio.h>

int asdf(int, int);
int choo(int, int);

int main() {
  int n, k;
  for(n=0;n<=16;++n) {
    printf("T<sub>%d</sub>(x)=", n);
    for(k=0;k<=n/2;++k) {
      printf("%s%d", k&&asdf(n, k)>=0?"+":"", asdf(n, k));
      if(n-2*k) printf("x");
      if(n-2*k>1) printf("<sup>%d</sup>", n-2*k);
      if(k==n/2) printf("<br>\n");
    }
  }
  return(0);
}

int tmp[1000];

int asdf(int n, int k) {
  int r, a;
  if(!n) return(1);
  if(!k) return(1<<(n-1));
  for(a=r=0;a<k;++a) r-=(asdf(n, a)*choo(n-2*a, k-a))>>(2*(k-a));
  return(r);
}

int choo(int n, int k) {
  int x, y;
  for(tmp[0]=y=1;y<=n;++y) for(tmp[x=y]=0;x;--x) tmp[x]+=tmp[x-1];
  return(tmp[k]);
}</pre>

----

:If we recursively define a function (the letter &xi; an arbitrary choice)
::<math>\xi(n,0)=\lceil 2^{n-1}\rceil</math>
::<math>\xi(n,k)=-\sum^{k-1}_{a=0}\frac{\xi(n,a)}{4^{k-a}}{n-2a\choose k-a}</math>
:then
::<math>T_n(x)=\sum^{\lfloor n/2\rfloor}_{k=0}\xi(n,k)x^{n-2k}</math>

==Maxima and minima==
It looks to me like the local maxima of the Chebyshev polynomials of the first kind all have value 1, and the local minima all have value -1. If so (I don't know enough to say whether it is actually true), then this is an interesting property that should be mentioned somewhere on the page. I'm guessing it probably pops right out of one of the definitions, but that's not transparent to a muggle like me. -[[User:Dmmaus|dmmaus]] 04:01, 15 August 2005 (UTC)

I think that follows from the definition:
:<math>T_n(x) = 
\left\{
\begin{matrix} 
& \cos(n\arccos(x))           & \mbox{ , } \ x \in [-1,1] \\
& \cosh(n \, \mathrm{arcosh}(x))         & \mbox{ , } \ x \ge 1      \\
& (-1)^n \cosh(n \, \mathrm{arcosh}(-x)) & \mbox{ , } \ x \le -1     \\
\end{matrix}\right.
</math> 

Obviously, for x in [-1, 1] since the behaviour is given by the cosine, the  local maxima and minima are 1 and -1. If one can prove that there are no maxima/minima outside this interval, then you assertion would follow. [[User:Oleg Alexandrov|Oleg Alexandrov]] 04:07, 15 August 2005 (UTC)

Ah, good point. I'm afraid my eyes glaze over when I see hyperbolic trig functions in the vicinity. :-)  Still, I think it'd be nice to mention this property explicitly in the article, since it's the standout physical feature of the graphs when you look at them. -[[User:Dmmaus|dmmaus]] 08:54, 15 August 2005 (UTC)
:: I don't know that much about these polynomials to attempt to write in the article. Let us hope somebody else will mention this fact. [[User:Oleg Alexandrov|Oleg Alexandrov]] 17:04, 16 August 2005 (UTC)

:::I've mentioned it.  
:::There's also an interesting paralel here, which is a bit OT.  Notice that for a degree n polynomial (with fixed leading coeff) to have a minimal maximum absolute value on an interval, it has to reach this maximal absolute value the highest number of times possible: n+1 times.  (Clearly a degree n polynomial can have at most n-1 places where its derivative is 0, so it's impossible that is has more than n+1 extremal point in an interval.)  This paralels up nicely with a general theorem in numerical analysis that the polynomial p of degree n (with no restriction on the leading coeff) that approximates a continuous function f in the sense that max|p-f| on an interval is minimal, |p-f| has to take this maximum n+1 times.  I'm not sure how this general theorem is stated exactly, and I also don't know if it has any connection to this.  &#x2013; [[User:b_jonas|b_jonas]] 21:45, 29 December 2005 (UTC)

:::: You have cited famous approximation theorem by Chebyshev (not later than 1857). And yes, extremal properties of polynomials of first kind (minimal C-norm) follow from this theorem immediately. [[User:Mir76|Mir76]] ([[User talk:Mir76|talk]]) 16:18, 16 August 2009 (UTC)

:::Let me say thanks to [[User:172.128.197.224]] for his correction of the statement in the second paragraph of the article.  &#x2013; [[User:b_jonas|b_jonas]] 18:30, 18 February 2006 (UTC)

== Optimal approximations to arbitrary functions ==

The theorem says that, if an n<sup>th</sup> degree polynomial P
approximates a function F on a closed interval in such a way that the
error function P(x)-F(x) has n+2 maxima, and those maxima have
alternating signs and the same absolute value (that is, P-F has maxima
+&epsilon;, -&epsilon;, +&epsilon;, ...), then that polynomial is '''locally'''
optimal.  That is, there is no other polynomial '''close to P''' that
has maximum error less than &epsilon;.  This locality situation is the
same as the situation in differential calculus: elementary calculus
problems involve finding the maximum value of a function.  One sets
the derivative to zero.  That gives a '''local''' maximum (or
minimum), but there might be other solutions that are far away.

I will outline the proof below.  I think this theorem, and its proof,
are interesting, but don't seem to be part of the standard maths
curriculum.  (Orthogonal polynomials in general seem to be a
mathematical orphan child.)  I would suggest that this proof ought to
be on a separate "proof" page.  WP is trying to minimize the number of
such pages, because WP isn't a textbook, but this would probably be a
good candidate.

The proof really requires a graph to motivate it.  I'm new to WP; it
should be made by someone who is comfortable making and uploading
graphs, such as the graphs on the Chebyshev page.

Here's the proof: Suppose P(x)-F(x) has n+2 maxima, alternating signs,
all with absolute value &epsilon;.  Suppose Q is another n<sup>th</sup>
degree polynomial very close to P, that is strictly better.
Superimpose tha graph of Q-F on that of P-F.  Since Q is close to P,
they will be about the same, but at the n+2 points where P-F is
maximal, Q-F must lie strictly inside P-F.  This is because the
maximal excursions of Q-F are &delta;, which is strictly less than
&epsilon;.  So (Q-F)-(P-F) is strictly nonzero at those n+2 points, with
alternating signs.  But (Q-F)-(P-F) = Q-P, an n<sup>th</sup> degree
polynomial.  Since it has n+2 alternating nonzero values, it must have
n+1 roots.  But n<sup>th</sup> degree polynomials can't do that.

Now here's an outline of why Chebyshev interpolation gets very close
to this optimum polynomial: If the Taylor series for a function
converges very rapidly (the way, for example, sin, cos, and
exponential do), the error from chopping off the series after some
finite number of terms will be close to the first term after the
cutoff.  That term will dominate all the later terms, and that term
plus all the later ones are of course the error function.  The same is
true if we use a Chebyshev series instead of a Taylor series.

Now if we expand F(x) in its Chebyshev series, and chop it off after
the T<sub>n</sub> term, we have an n<sup>th</sup> degree polynomial.
The error is dominated by the T<sub>n+1</sub> term, so it looks like
T<sub>n+1</sub>.  T<sub>n+1</sub> is level, with n+2 maxima, with
alternating signs.  This means the error function is
'''approximately''' level, and hence, by the theorem above, this
polynomial is approximately optimal.

If one wants a truly optimal polynomial, one "tweaks" the polynomial P
so that its error excursions are truly level.  Remes' algorithm does
this.

Should there be a separate WP page for this stuff, linked from Chebyshev?
What should it be called?
[[User:William Ackerman|William Ackerman]] 18:46, 21 February 2006 (UTC)

:Thanks for stating the theorem.  &#x2013; [[User:b_jonas|b_jonas]] 21:13, 21 February 2006 (UTC)

== The relationship ==

I removed the following: 

::Starting with T0( cos u ) = 1 and T1( cos u ) = cos u, one quickly calculates that cos 2u = 2 cos^2 u - 1, cos 3u = 4 cos^2 u - 3 cos u, cos 4u = 8 cos^4 u - 8 cos^2 u + 1, etc. By setting u = 0, whence cos u = 1, one can immediately detect that both sides of each equation evaluate to unity, which is a good clue that the relationship may be correct&#151;which, of course, it is.

What does "the relationship may be correct" mean? which relationship? Not <math>T_n(\cos\theta)=\cos(n\theta)</math> because thats a definition. It may show that the trignometric identities are true, but thats not the point of the article. [[User:PAR|PAR]] 13:27, 11 August 2006 (UTC)

== reasoning not correct ==

I removed the following part because it is nonsense: 

 From reasoning similar to that above, one can develop a [[Closed-form expression|closed-form]] generating formula for Chebyshev polynomials of the first kind:

:<math>
\cos(n \theta)=\frac{e^{i n \theta}+e^{-i n \theta}}{2}=\frac{(e^{i \theta})^n+(e^{i \theta})^{-n}}{2} \,\!
</math>

which, combined with [[DeMoivre's formula]]: '''THIS IS FALSE:'''

:<math>
\! e^{i \theta}=\cos\theta+i \sin\theta=\cos\theta+i \sqrt{1-\cos^2\theta}=\cos\theta+\sqrt{\cos^2\theta-1} \,\!
</math>

yields:

:<math>
\cos(n \theta)=\frac{\left(\cos\theta+ \sqrt{\cos^2\theta-1}\right)^n+\left(\cos\theta+ \sqrt{\cos^2\theta-1}\,\right)^{-n}}{2} \,\!
</math>

which, of course, is far more expedient for determining the cosine of ''N'' times a given angle than is cranking through almost ''N'' rounds of the recursive generator calculation. Finally, if we replace <math>\cos(\theta)</math> with ''x'', we can alternatively write:

:<math>
 T_n(x)=\frac{\left(x+ \sqrt{x^2-1}\right)^n+\left(x+ \sqrt{x^2-1}\right)^{-n}}{2}.
</math>

==Linear operators==
Should someone write a section about these as the eigenfunctions of a second order linear differential operator (which is symmetric wrt the inner product <math>(f,g)= \int_{-1}^{1} f(x)g(x)\frac{1}{\sqrt{1-x^2}}\,dx </math>)? I would, but I'm not really comfortable with the theory yet... [[User:Marbini|Marbini]] ([[User talk:Marbini|talk]]) 17:54, 2 April 2008 (UTC)

== Factoring ==

Can anything of interest be said about factorization of Chebyshev polynomials? [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 00:04, 9 December 2008 (UTC)

:Apparently. Googling for "factoring chebyshev" turned up [http://adm.lnpu.edu.ua/downloads/issues/2008/N2/adm-n2-4.pdf this] and [http://icm.mcs.kent.edu/reports/1999/chebpol.pdf this] which seem to at least contain some pointers. [[User:Fredrik|Fredrik Johansson]] 00:15, 9 December 2008 (UTC)

So there's really not very much.....maybe.  There is a sense in which the [[spread polynomials]] ''S''<sub>''n''</sub>, satisfying
: <math> S_n(\sin^2\theta) = \sin^2(n\theta)\,</math>
are trivially equivalent to the Chebyshev polynomials ''T''<sub>''n''</sub>, satisfying
: <math> T_n(\cos\theta) = \cos(n\theta).\,</math>
But the article titled [[spread polynomials]] has some material on factorization, and maybe there the story is different.  Or maybe not? [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 21:36, 10 March 2009 (UTC)

Some interest in factoring the Chebyshev T-polynomials stems from reduction of cosines to rational numbers as [http://www.mpia-hd.mpg.de/~mathar/public/mathar20060529.pdf PDF here]. - [[User:R. J. Mathar|R. J. Mathar]] ([[User talk:R. J. Mathar|talk]]) 19:35, 12 March 2014 (UTC)

A few remarks about divisibility of Chebyshev polynomials by other Chebyshev polynomials have now been added in the section [[Chebyshev polynomials#Composition and divisibility properties|Composition and divisibility properties]]. Perhaps more could be said, but I wanted to keep it simple, and also didn't feel qualified to say more. [[User:Will Orrick|Will Orrick]] ([[User talk:Will Orrick|talk]]) 23:24, 8 January 2022 (UTC)

== Explicit formulas ==

Can someone provide sources for the given explicit formulas, or at least some hints on how to obtain them?

I found some other formulas in Mason, J. C. "Chebyshev polynomials", 2003. CRC Press, 2003., pp.35-36 and in Snyder, M. A. "Chebyshev methods in numerical approximation", Englewood Cliffs, NJ, USA, 1966., p. 14.

In the first book, there are some other references to Rivlin (1974) and Clenshaw (1962). [[User:Mstempin|Mstempin]] ([[User talk:Mstempin|talk]]) 21:23, 3 August 2009 (UTC)

:After expanding formula 
::<math>T_{n}(x)=\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor}{n \choose 2k}x^{n-2k}(x^2-1)^{k}</math>
:for n=8 i made conjecture that 
::<math>T_{n}(x)=\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor}\sum_{m=k}^{\lfloor\frac{n}{2}\rfloor}(-1)^k {n \choose 2m}\cdot {m \choose k} x^{n-2k}</math>
:This conjecture should be proven and in my opinion induction should be enough
:For the sum 
::<math>\sum_{m=k}^{\lfloor\frac{n}{2}\rfloor} {n \choose 2m}\cdot {m \choose k}</math>
: i found with help of Wolfram Alpha only piecewise definition 
:: <math>\sum_{m=k}^{\lfloor\frac{n}{2}\rfloor} {n \choose 2m}\cdot {m \choose k} = \begin{cases}1 \qquad n=0 \\ \frac{n}{2}\cdot\frac{1}{n-k}\cdot {n-k \choose k}\cdot 2^{n-2k} \qquad n > 0\end{cases}</math>  <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/188.47.53.242|188.47.53.242]] ([[User talk:188.47.53.242#top|talk]]) 11:56, 4 July 2023 (UTC)</small> <!--Autosigned by SineBot-->

== History ==

I believe Chebyshev developed these polynomials while attempting to solve the problem of the [[inversor]], a mechanical linkage that converts circular motion into linear motion. [[User:DonPMitchell|DonPMitchell]] ([[User talk:DonPMitchell|talk]]) 03:42, 13 March 2010 (UTC)

== Dickson polynomial ==
{{Discussion top|1=The result of this discussion was '''no consensus'''. [[User:NukeofEarl|NukeofEarl]] ([[User talk:NukeofEarl|talk]]) 16:24, 30 October 2014 (UTC)}}
I would like to propose merging [[Dickson polynomial]] into [[Chebyshev polynomials]].  All the properties, except the analysis over finite fields, is already here.  — [[User:Arthur Rubin|Arthur Rubin]]  [[User talk:Arthur Rubin|(talk)]] 00:27, 21 September 2011 (UTC)

:''Dickson polynomials are mainly studied over finite fields, when they are not equivalent to Chebyshev polynomials.''
:It appears in their application, these two polynomials groups do really differ, and I wouldn't support a merge... [[User:Gryllida|Gryllida]] ([[User talk:Gryllida|talk]]) 22:16, 2 January 2014 (UTC)
{{Discussion bottom}}

== Proposed section to merge: Spread polynomials and some other Chebyshev-linked Polynomials ==
[[spread polynomials]] are in a sense equivalent to the Chebyshev polynomials of the first kind, but enable one to avoid square roots and conventional trigonometric functions in certain contexts, notably in [[rational trigonometry]].
:'''[[Dickson polynomials]]'''<ref> [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.71.8553&rep=rep1&type=pdf]
</ref><ref> [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.71.8553&rep=rep1&type=pdf]
</ref><ref> [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.34.5317&rep=rep1&type=pdf]</ref><ref> [http://www.math.okayama-u.ac.jp/mjou/mjou1-46/mjou_pdf/mjou_33/mjou_33_001.pdf]</ref> ,''D''<sub>''n''</sub>(''x''), are  equivalent to [[Chebyshev polynomial]]s ''T''<sub>''n''</sub>(''x''), with a slight and trivial change of variable:
:<math> D_n(2x) = 2 T_n(x) \,</math>

Dickson polynomials are  given by:

:<math>D_n(x)=\sum_{p=0}^{\lfloor n/2\rfloor}\frac{n}{n-p} \binom{n-p}{p} (-x^{n-2p}. </math>

The first few Dickson polynomials  are

:<math> D_0(x) = 2 \,</math>

:<math> D_1(x) = x \,</math>

:<math> D_2(x) = x^2 - 2 \,</math>

:<math> D_3(x) = x^3 - 3x \,</math>

:<math> D_4(x) = x^4 - 4x^2 + 2 \,</math>
[[Boubaker polynomials]] are also linked to Chebyshev polynomials. they can be defined by the [[recurrence relation]]<ref>O.D. Oyodum, O.B. Awojoyogbe, M.K. Dada, J.N. Magnuson, [http://runners.ritsumei.ac.jp/cgi-bin/swets/hold-query-e?mode=1&key=&idxno=29124246 Comment on "Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition" by K. Boubaker, A. Chaouachi, M. Amlouk and H. Bouzouita. On the earliest definition of the Boubaker polynomials], ''Eur. Phys. J. Appl. Phys.,'' Volume 46, pp. 2120-21202.</ref><ref>Paul Barry and Aoife Hennessy, Journal of Integer Sequences (JIS),Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, Chapter 6: '''''The Boubaker polynomials '''''  [http://www.emis.ams.org/journals/JIS/VOL13/Barry5/barry96s.pdf]</ref><ref>   A. S.  Kumar  , An analytical solution to applied mathematics-related Love's equation using the ‘’’Boubaker polynomials’’’  expansion scheme| journal=International Journal of the Franklin Institute (elsevier) [http://cat.inist.fr/?aModele=afficheN&cpsidt=23388093]</ref><ref> Encyclopedia of Physics Research, Chapter 21: Cryogenics Vessels Thermal Profilng Using the Boubaker Polynomials, Editors: Nancy B. Devins and Jillian P. Ramos,   [https://www.novapublishers.com/catalog/product_info.php?products_id=26337&osCsid=] </ref>:

:<math>B_n(x)= \begin{cases}
1, & \mbox{if } n =  0, \\
x, & \mbox{if } n =  1, \\
x B_{n - 1}(x) - B_{n - 2}(x),& \mbox{if } n \geq 3.
\end{cases},</math>

[[Lucas polynomials]] ''L''<sub>''n''</sub>(''x'')<ref>[http://mathworld.wolfram.com/LucasPolynomial.html]</ref> are a [[polynomial sequence]] defined by the following Chebyshev-like recurrence relation:

:<math>L_m(x) = \begin{cases}
2, & \mbox{if } m = 0 \\
x, & \mbox{if } m = 1 \\
x L_{m - 1}(x) + L_{m - 2}(x), & \mbox{if } m \geq 2.
\end{cases}</math>

{{reflist-talk}}

== Possible corrections ==
=== Section on Orthogonality and Example 1 has errors===

The session on Orthogonality has a somewhat major issue, namely that the discrete orthogonality condition as written is wrong. The discrete orthogonality condition neglects to mention that the formula given is only valid for <math>0\leq i,j \leq N-1</math>. For other values of <math>i,j</math> the aliasing condition gives alternating sign "tiled" versions of what's currently written.

Also, in Basis Set, Example 1, the closed-form Chebyshev coefficients <math>a_n</math> for Log(1+x) is given correctly. However, immediately afterward it is written that these same coefficients <math>a_n</math> can alternately be computed via the discrete Gauss-Lobatto sum. This is false, as is readily verified numerically. The <math>a_n</math> in the Gauss-Lobatto discretization are the aliasing-folded versions of the <math>a_n</math> computed in the continuous integral case, and are not the same. To a newcomer who does not know this already, this is completely non-obvious, especially considering that the same symbol <math>a_n</math> is used for both. I suggest that different symbols should be used for the two cases to emphasize the fact that the two quantities are different.  <small><span class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:MathDoobler|MathDoobler]] ([[User talk:MathDoobler|talk]] • [[Special:Contributions/MathDoobler|contribs]]) 04:13, 3 January 2014 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

=== Wrong discrete orthogonality ===
There is a very wrong formula in the Orthogonality section. The discrete orthogonality shouldn't have the weight that appears in the continuous orthogonality condition. See for example Eq. (3.30) of https://www.siam.org/books/ot99/OT99SampleChapter.pdf
Revision of 14 March 2016 should be undone because wrong!
[[User:Pliskin14|Pliskin14]] ([[User talk:Pliskin14|talk]]) 09:21, 12 May 2016 (UTC)

== Third and Fourth Kind ==
Is there any reason why the Chebyshev polynomials V_n(x) and W_n(x) of the third and fourth kind are not mentioned (at least: defined) in the article? [[User:R. J. Mathar|R. J. Mathar]] ([[User talk:R. J. Mathar|talk]]) 19:29, 12 March 2014 (UTC)
:''I'' hadn't heard of them.  Got a reference which is uses them?  — [[User:Arthur Rubin|Arthur Rubin]]  [[User talk:Arthur Rubin|(talk)]] 16:27, 18 March 2014 (UTC)
: For example 
# {{cite book|author=J. C. Mason|section=1.2.3|title=Chebyshev Polynomials|year=2010|publisher=CRC Press}}
# {{cite news |doi=10.1016/0377-0427(93)90148-5|author =J. C. Mason|title=Chebyshev polynomials of the second, third and fourth kinds in approximation...}}
# {{cite news |author=J. C. Mason, G. H. Elliott|title=Constrained near-minimax approximation...|journal=Num. Alg. | year=1995|pages=39-54|number=9|doi=10.1007/BF02143926}}
# {{cite news |author=S. E. Notaris|title=Interpolatory quadrature..|doi=10.1016/S0377-0427(97)00018-6}}
# {{cite news |doi=10.1016/j.amc.2007.09.018|title=A survey on third and fourthkind...|year=2008|journal=Appl. Math Comput.}}
# {{cite news |author=G. Heinig|title=Chebyshev-hankel matrices...|year=2001|pages=181-196|journal=Lin. Alg. Applic.|doi=10.1016/S0024-3795(00)00300-1}}
and many, many others [[User:R. J. Mathar|R. J. Mathar]] ([[User talk:R. J. Mathar|talk]]) 17:25, 1 May 2014 (UTC)

In response to my earlier edit being reverted, p_1 as listed are not orthogonal to the respective weight functions despite being consistent with the definitions. While definitions for V and W are sometimes interchanged, the ones used here are consistent with Mason and Handscomb's 2003 book (definition 1.3), but the weight functions are not (section 4.2.2). In regards to this page, I believe that the weight functions for V and W need to be switched.  <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Youboo4|Youboo4]] ([[User talk:Youboo4#top|talk]] • [[Special:Contributions/Youboo4|contribs]]) 21:12, 21 July 2022 (UTC)</span> <!--Autosigned by SineBot-->

:I now agree that the weights were incorrect and have switched them. Do you agree that everything else is consistent now? [[User:Will Orrick|Will Orrick]] ([[User talk:Will Orrick|talk]]) 22:15, 21 July 2022 (UTC)
::Yes, I think that was the only issue. [[User:Youboo4|Youboo4]] ([[User talk:Youboo4|talk]]) 23:45, 21 July 2022 (UTC)

Regarding the 24 May 2024 edits and reversions: it seems to have escaped the participants' notice that Chebyshev polynomials of the third and fourth kind were already in the article. My suggestion would be to roll the article back to its state prior to 24 May 2024 and to have a discussion before changing the article in such a drastic way. (As it is now, the article is only partially rolled back.) [[User:Will Orrick|Will Orrick]] ([[User talk:Will Orrick|talk]]) 06:50, 26 May 2024 (UTC)

:I rolled back all the recent edits that introduced polynomials of the third and fourth kind early in the article. I'm willing to entertain contrary opinions, but my feeling is that 99% of readers coming to this article are here to learn about polynomials of the first or second kind, which have a history going back to Viète and are ubiquitous in mathematics and science. Polynomials of the third and fourth kinds, while implicit in the work of Jacobi, and having been studied in the aerospace community for more than half a century, have only started to be called by that name in the past few decades and have a much smaller literature. I feel that introducing them early in the article gives them undue weight.

:Furthermore, it is untenable to have an article that opens by stating that there are two sets of polynomials, then immediately introduces four of them, but then goes back mostly to talking as if there are only two sets, only to reintroduce the other two sets at the very end of the article as if they hadn't been mentioned before. This is what we had prior to the rollback. If treating Chebyshev polynomials as four sets of polynomials is the right thing to do from the outset, then the article needs to maintain that point of view throughout.

:I'm not saying that such a change would need to take place in a single edit—that would be impractical—but anyone proposing to make that change needs to provide a rationale that achieves broad support from other editors and needs to outline a plan for carrying it out. [[User:Will Orrick|Will Orrick]] ([[User talk:Will Orrick|talk]]) 16:09, 26 May 2024 (UTC)
::I think it's possibly worth mentioning in the lead section, but perhaps not worth elaborating too much up front. My impression is that these are mostly used in the context of fast transforms, where a polynomial of type X is broken into sub-transforms of types X, Y, Z (precisely which ones depends on X and on the degree). –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 07:24, 27 May 2024 (UTC)

:::The questions of whether and where to mention the third and fourth kinds comes down to how widespread usage of those terms is. I was the person who added mention of them to the article back in January 2022. I did that because R. J. Mathar's comment on this Talk page persuaded me that usage of the terms was common enough that we couldn't ignore it. On the other hand, my judgement was that this usage was restricted to certain specialties and was relatively new, so I restricted the mention to a late section of the article, "Families of polynomials related to Chebyshev polynomials".

:::Whenever terminology with longstanding use is adopted for a new purpose, the new thing is piggybacking on the fame of the older one. Sometimes the new usage catches on; sometimes it doesn't; sometimes it catches on in some quarters but not in others. I find it hard to judge whether the applications you mention are important enough to mention in the lead of this article. [[User:Will Orrick|Will Orrick]] ([[User talk:Will Orrick|talk]]) 11:46, 28 May 2024 (UTC)
::::Sure, I agree. I might even focus on the "first-kind" ''T'' polynomials, which are what people mean when they say "Chebyshev polynomials" without qualification, and seem by far the most important to discuss, and then lump the second, third, and fourth-kind spin-offs into a later section. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 15:59, 28 May 2024 (UTC)

:::::This is valuable input, and shows that there is a wide variety of different views. It's hard to get many editors to weigh in on issues like this, but I hope we will hear from others.

:::::I think I learned this topic from volume II of Bateman (1953), or maybe from Abramowitz and Stegun (1964), both of which use the language "first kind", "second kind" and deal with two sets of polynomials from the start. It's also interesting that polynomials essentially equivalent to {{mvar|T(x)}} and {{mvar|U(x)}} already appear together in Viète's book (1646). But it seems you may be right that this "first kind", "second kind" vocabulary is of relatively recent vintage. I find very little evidence of this language prior to Bateman. Regarding terminology, Bateman (p. 183) has this to say,

::::::"Sometimes (especially in the French literature) orthogonal polynomials in general are called Tchebichef polynomials. There are also several special systems of orthogonal polynomials called Tchebichef polynomials. In this chapter we shall reserve the name Tchebichef polynomials of the first and second kind for the suitably standardized orthogonal polynomials associated with
:::::::<math>a=-1,\quad b=1,\quad w(x)=(1-x^2)^{\mp1/2},\quad X=1-x^2</math>."

:::::A quick search yielded articles on Chebyshev polynomials of the fifth, sixth, seventh, and eighth kinds, so it looks like this is never going to end. [[User:Will Orrick|Will Orrick]] ([[User talk:Will Orrick|talk]]) 13:19, 29 May 2024 (UTC)
::::::It's probably worth going with whatever is used in widely cited dedicated sources, e.g. Rivlin (1974) or Mason & Handscomb (2002). I don't feel particularly motivated to do a survey right now though. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 14:57, 29 May 2024 (UTC)
::::::Just my perspective, but I've read through most of this page multiple times in the capacity of my research (computer science / applied math), and not yet encountered a reason to use Chebyshev polynomials of the third & fourth kind. My impression is that the first kind is by far the most important, and expressions involving the second kind come up often when analyzing the first kind, so it is important by proxy. I haven't seen any use for these other polynomials in the areas I work in, and to my knowledge it's not even common to find it in approximation theory textbooks. So, keeping weight off of the third and fourth kind makes sense to me. [[User:Fawly|Fawly]] ([[User talk:Fawly|talk]]) 20:26, 29 May 2024 (UTC)

::::::So I guess the Mason of Mason and Handscomb is the Mason that wrote all those articles on 3rd and 4th kind polynomials mentioned by R. J. Mathar in the opening comment of this thread. It's therefore perhaps not surprising that the blurb for the book says that it treats "all four kinds" of Chebyshev polynomials, and, indeed, the table of contents bears that out.
::::::Rivlin had a second edition in 1990, which was just about the time that the 3rd kind/4th kind terminology was being introduced, so one wouldn't expect to find it in that book. From what I can see of the first chapter, "the" Chebyshev polynomials are the {{math|''T''(''x'')}}. But Rivlin does introduce the second kind polynomials as early as page 7.
::::::The NASA document by Desmarais and Bland, cited in the bibliography discusses the original motivation for looking at the 3rd and 4th kind polynomials.
::::::I don't work in approximation theory, so I don't have much of an opinion. Chebyshev polynomials, of both of the first two kinds, come up from time to time in my work in mathematical physics, combinatorics, and number theory, but that's about it. [[User:Will Orrick|Will Orrick]] ([[User talk:Will Orrick|talk]]) 13:18, 30 May 2024 (UTC)
:::::::Fox & Parker (1968) also seems to focus on the T polynomials. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 02:30, 31 May 2024 (UTC)
:::::::Boyd (1989) ''Chebyshev and Fourier Spectral Methods'' focuses on the ''T'' polynomials, but also has two appendices covering the "Chebyshev polynomials: {{tmath|T_n(x)}}" and the "Chebyshev polynomials of the second kind: {{tmath|U_n(x)}}", respectively. Davis (1963) ''Interpolation and Approximation'' also focuses on the ''T'' for Tschebyscheff polynomials, and when he later introduces the U polynomials, describes the two types as "Tschebyscheff polynomials (of the first kind)" and "Tschebyscheff polynomials of the second kind". –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 02:47, 31 May 2024 (UTC)
:::::::Trefethen (2013) [https://people.maths.ox.ac.uk/trefethen/ATAP/ ''Approximation Theory and Approximation Practice''] focuses on the ''T'' polynomials. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 02:50, 31 May 2024 (UTC)
:::::::Cheney (1966) ''Introduction to Approximation Theory'' focuses on the "Tchebycheff polynomials" ''T'', but introduces the "Tchebycheff polynomials of the second kind" ''U'' in an exercise, and then uses them a couple more times toward the end of the book. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 03:22, 31 May 2024 (UTC)

== Series coefficients found using Gauss–Lobatto zeros are approximate ==

I edited the section "As a basis set," Example 1, to indicate that the coefficients found by exploiting the discrete orthogonality property and the Gauss–Lobatto zeros give approximate results. This is discussed in Numerical Recipes (which is about my level of expertise here).

Also, I believe that the "exact" results in the same example for n > 0 are wrong. I don't have a correct form but I have a short Mathematica notebook which compares these exact coefficients with the approximate ones using a large ''N'' and they aren't even close.  <small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/97.117.195.123|97.117.195.123]] ([[User talk:97.117.195.123|talk]]) 10:33, 22 September 2015 (UTC)</small><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

Oh--sorry. The comment not far above this, "Section on Orthogonality and Example 1 has errors," deals with the same subject.  <small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/97.117.195.123|97.117.195.123]] ([[User talk:97.117.195.123|talk]]) 10:35, 22 September 2015 (UTC)</small><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

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== Proposed remove mention of spread polynomials ==
They seem to be part of the research program of one author and to have not caught on more widely.  Googling "spread polynomials" and "Chebyshev polynomials" gives me 1.5k vs 364k hits. Spread polynomials are introduced in a book which is not widely cited apart from by its author.  The work may be good but that shouldn't be wikipedia's job to evaluate. This [https://mathoverflow.net/questions/47520/spread-polynomials mathoverflow thread] suggests that they are basically equivalent, and that it's enough to just describe shifted Chebyshev polynomials. I propose removing all mention of spread polynomials from the article. The only reason to keep them, I think, would be if we find the approach to geometry in the book "Divine Proportions..." to be an important enough application  of Chebyshev polynomials to mention in the main article.  But Chebyshev polynomials have many many important applications I'd mention first, like their role in filter design, linear algebra algorithms, etc. [[User:Aram.harrow|Aram.harrow]] ([[User talk:Aram.harrow|talk]]) 19:17, 23 March 2021 (UTC)

update: I've edited [https://en.wikipedia.org/w/index.php?title=Chebyshev_polynomials&diff=1014732963&oldid=1012534150 the article] to remove them. [[User:Aram.harrow|Aram.harrow]] ([[User talk:Aram.harrow|talk]]) 20:02, 28 March 2021 (UTC)

== Generalized Chebyshev polynomials ==

The term "generalized Chebyshev polynomials" appears to have a variety of uses: among others, it is used to refer to  [[Dessin d'enfant#Trees and Shabat polynomials|Shabat polynomials]], to certain two-variable or multi-variable generalizations, and to generalizations to unions of disjoint intervals. As used in the section "Generalized Chebyshev polynomials", however, it appears to refer to objects that are not actually polynomials most of the time. I have not been able locate any sources attesting to this use of the term. Would anyone be able to help? I have placed a "dubious" tag on the section. [[User:Will Orrick|Will Orrick]] ([[User talk:Will Orrick|talk]]) 05:41, 9 January 2022 (UTC)

:I have removed the section. It seems unlikely that this is a common use of the term "generalized Chebyshev polynomials" if extensive searching doesn't turn up any examples of the term being used in this way. [[User:Will Orrick|Will Orrick]] ([[User talk:Will Orrick|talk]]) 03:40, 15 January 2022 (UTC)

==Hyperbolic trigonometric functions; romanization of "Chebyshev"==
 
Your argumet of cosh is  purely imaginary assuming both x,t in R in your exponential generating function of T_{n}  <br/>
Would be better to use trig cos instead<br>
By the way this guy was named Chebyshov because over last e there is diaeresis which Russians usually dont write <br/>
Russians usually dont write it but diareresis is still here <br>  <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/188.47.27.31|188.47.27.31]] ([[User talk:188.47.27.31#top|talk]]) 17:57, 29 May 2023 (UTC)</small> <!--Autosigned by SineBot-->

:Wikipedia's [[Pafnuty Chebyshev|biographical article]] on Chebyshev mentions the romanization issue. Unfortunately, the current spelling is well established in the mathematics literature, and Wikipedia needs to reflect what's in the majority of authoritative sources. [[User:Will Orrick|Will Orrick]] ([[User talk:Will Orrick|talk]]) 01:31, 30 May 2023 (UTC)
::Anyone interested in Chebyshev and romanization might enjoy Philip Davis's [https://archive.org/details/threadmathematic00davi ''The Thread'']. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 16:09, 28 May 2024 (UTC)

Yes but current transcription leads to misreading this name Many people misread this name  <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/188.47.53.242|188.47.53.242]] ([[User talk:188.47.53.242#top|talk]]) 23:38, 7 July 2023 (UTC)</small> <!--Autosigned by SineBot-->

== Incorrect integration formula for n=0 ==

In the integration "The last formula can be further manipulated to express the integral of Tn as a function of Chebyshev polynomials of the first kind only:" the derivation is incorrect for n=0. It uses T_1 T_n = 1/2 (T_{n+1} + T_{n-1}). But according to the below section "Products of Chebyshev polynomials" the T_{n-1} should be T_{|n-1|}. So the formula is incorrect for n=0. [[Special:Contributions/100.7.38.140|100.7.38.140]] ([[User talk:100.7.38.140|talk]]) 19:41, 26 August 2023 (UTC)

== Original publication ==

In the reference it is written "Chebyshev polynomials were first presented in: (1854)". That is simply wrong, for it was in 1853. Then there is a topic of a "journal article". Why was the first publication not a journal article?.. Why was the second one that one?
[[User:Yaroslav Nikitenko|Yaroslav Nikitenko]] ([[User talk:Yaroslav Nikitenko|talk]]) 17:11, 26 March 2025 (UTC)

:Chebyshev presented his paper in January 1853 to the Academy of St. Petersburg to be published in their journal (''Memoirs of Foreign Scholars presented to the Academy of Saint Petersburg'') which published in 1854. It was also separately published as a pamphlet/short book. You can see in your link that the book says that the paper was "extracted from" the journal. In this kind of situation the primary reference is the paper in the context of the journal, not the extract published separately. If you really want you can put something like "orig-date=Presented 1853" in the first template, but I wouldn't bother; I don't think readers are hung up on the precise details of 19th century academic publication timing. (This is much less dramatic than many Euler papers which were presented in one year but only finally published years or even decades afterward.)  –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 17:31, 26 March 2025 (UTC)
::Oh, thanks, this is really interesting. I am really confused why it was *first time* published in "Memoirs of Foreign Scholars" if Chebyshev was Russian. Yes, I looked at the book and didn't understand well why it was "extracted" in 1853 from a later publication. Do you have any source material to that? Yes, I would add that it was "originally presented in 1853". The whole situation is not very clear for me though. - [[User:Yaroslav Nikitenko|Yaroslav Nikitenko]] ([[User talk:Yaroslav Nikitenko|talk]]) 09:20, 27 March 2025 (UTC)
:::Sometimes if there's a paper that seems like it will have broader interest it will also be published separately as a book or pamphlet. In this case maybe there was also a goal of getting a copy to people a bit faster? It takes a while to prepare a whole journal volume for publication, and took even longer a couple centuries ago. This journal volume was not published until November 1854. {{tq|"... if Chebyshev was Russian"}} – That's just the name of the journal; it apparently doesn't mean that only foreigners were allowed to publish there. Shrug. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 09:37, 27 March 2025 (UTC)
::::Yes, it makes sense. You've changed the reference to "Chebyshev first presented his eponymous polynomials in a paper read before the St. Petersburg Academy in 1853" - I wonder, what was the source of that information, that he actually read his paper before the Academy? How do you know that? Are you a historian in mathematics? Sadly I couldn't find any information about that "foreigners" journal. In any case that is not that important. And thank you for the other links. - [[User:Yaroslav Nikitenko|Yaroslav Nikitenko]] ([[User talk:Yaroslav Nikitenko|talk]]) 19:17, 27 March 2025 (UTC)
:::::The title page of the paper says "Lu le 28 janvier 1853". If you want a secondary source, §5 of {{doi|10.1006/jath.1998.3289}} begins with: "Chebyshev's first work on approximation, namely his paper "Théorie des mécanismes connus sous le nom de parallélogrammes", was read to the St. Petersburg Academy on January 28, 1853 and published in 1854." {{pb}} In my comment above I translated the name of the journal from French, ''Mémoires des Savants étrangers présentés à l'Académie de Saint-Pétersbourg''. –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 20:13, 27 March 2025 (UTC)

== History ==

It would be good to add a history section (people are generally [https://mathoverflow.net/questions/249480/what-is-the-story-behind-the-chebyshev-polynomials interested]). It is said that Chebyshev polynomials of the second kind were introduced by his students (I don't have a good reference though).

I've also found that some of the Chebyshev polynomials were known at least in the [[arxiv:2203.10955|16th century]].

The data of the "first publication" is also not very clear and is being discussed in a separate topic. - [[User:Yaroslav Nikitenko|Yaroslav Nikitenko]] ([[User talk:Yaroslav Nikitenko|talk]]) 09:26, 27 March 2025 (UTC)

:Perhaps start with {{doi|10.1142/9789814360296_0030}}, {{doi|10.1006/jath.1998.3289}}, or {{doi|10.1006/jath.2000.3476}}, or perhaps take a look at the references in {{doi|10.1090/qam/58024}}. (These turned up in a quick search.) –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 09:59, 27 March 2025 (UTC)