Talk:Chebyshev polynomials

message: The theorem says that, if an nth 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
+ε, -ε, +ε, ...), then that polynomial is locally
optimal.  That is, there is no other polynomial close to P that
has maximum error less than ε.  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 ε.  Suppose Q is another nth
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 δ, which is strictly less than
ε.  So (Q-F)-(P-F) is strictly nonzero at those n+2 points, with
alternating signs.  But (Q-F)-(P-F) = Q-P, an nth degree
polynomial.  Since it has n+2 alternating nonzero values, it must have
n+1 roots.  But nth 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 Tn term, we have an nth degree polynomial.
The error is dominated by the Tn+1 term, so it looks like
Tn+1.  Tn+1 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?
William Ackerman 18:46, 21 February 2006 (UTC)
replies: text: Thanks for stating the theorem.  – b_jonas 21:13, 21 February 2006 (UTC)
thread_title:  Optimal approximations to arbitrary functions 

message: 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—which, of course, it is.
What does "the relationship may be correct" mean? which relationship? Not T_n(\cos\theta)=\cos(n\theta) because thats a definition. It may show that the trignometric identities are true, but thats not the point of the article. PAR 13:27, 11 August 2006 (UTC)
thread_title:  The relationship 

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

message: 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. DonPMitchell (talk) 03:42, 13 March 2010 (UTC)
thread_title:  History 

message: === 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 0\leq i,j \leq N-1. For other values of i,j the aliasing condition gives alternating sign "tiled" versions of what's currently written.
Also, in Basis Set, Example 1, the closed-form Chebyshev coefficients a_n for Log(1+x) is given correctly. However, immediately afterward it is written that these same coefficients a_n can alternately be computed via the discrete Gauss-Lobatto sum. This is false, as is readily verified numerically. The a_n in the Gauss-Lobatto discretization are the aliasing-folded versions of the a_n 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 a_n 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.  — Preceding unsigned comment added by MathDoobler (talk • contribs) 04:13, 3 January 2014 (UTC)
thread_title:  Possible corrections 

message: 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 0\leq i,j \leq N-1. For other values of i,j the aliasing condition gives alternating sign "tiled" versions of what's currently written.
Also, in Basis Set, Example 1, the closed-form Chebyshev coefficients a_n for Log(1+x) is given correctly. However, immediately afterward it is written that these same coefficients a_n can alternately be computed via the discrete Gauss-Lobatto sum. This is false, as is readily verified numerically. The a_n in the Gauss-Lobatto discretization are the aliasing-folded versions of the a_n 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 a_n 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.  — Preceding unsigned comment added by MathDoobler (talk • contribs) 04:13, 3 January 2014 (UTC)
thread_title:  Section on Orthogonality and Example 1 has errors

message: 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!
Pliskin14 (talk) 09:21, 12 May 2016 (UTC)
thread_title:  Wrong discrete orthogonality 

message: 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.  — Preceding unsigned comment added by 97.117.195.123 (talk) 10:33, 22 September 2015 (UTC)
thread_title:  Series coefficients found using Gauss–Lobatto zeros are approximate 

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thread_title:  External links modified 

message: 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 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. Aram.harrow (talk) 19:17, 23 March 2021 (UTC)
thread_title:  Proposed remove mention of spread polynomials 

message: The term "generalized Chebyshev polynomials" appears to have a variety of uses: among others, it is used to refer to  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. Will Orrick (talk) 05:41, 9 January 2022 (UTC)
replies: text: 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. Will Orrick (talk) 03:40, 15 January 2022 (UTC)
thread_title:  Generalized Chebyshev polynomials 

message: Your argumet of cosh is  purely imaginary assuming both x,t in R in your exponential generating function of T_{n}  
Would be better to use trig cos instead
By the way this guy was named Chebyshov because over last e there is diaeresis which Russians usually dont write 
Russians usually dont write it but diareresis is still here   — Preceding unsigned comment added by 188.47.27.31 (talk) 17:57, 29 May 2023 (UTC)
replies: text: Wikipedia's 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. Will Orrick (talk) 01:31, 30 May 2023 (UTC)
replies: text: Anyone interested in Chebyshev and romanization might enjoy Philip Davis's The Thread. –jacobolus (t) 16:09, 28 May 2024 (UTC)
thread_title: Hyperbolic trigonometric functions; romanization of "Chebyshev"

message: 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. 100.7.38.140 (talk) 19:41, 26 August 2023 (UTC)
thread_title:  Incorrect integration formula for n=0