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The Chebyshev polynomials, named after Pafnuty Chebyshev,[1] are a sequence of polynomials related to the trigonometric multi-angle formulae.
We usually distinguish between
Chebyshev polynomials of the first kind, denoted Tn and are closely related to and
Chebyshev polynomials of the second kind, denoted Un which are closely related to
The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff (French) or Tschebyschow (German).
The Chebyshev polynomials Tn or Unare polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a 'polynomial sequence'.
If approximating some function on the interval one can use a polynomial approximation like:
to approximate its values. Because is between 0 and 1 the terms get smaller from left to right. With a sufficiently large number of terms, usually more than we have here, we can typically get very good approximations to well behaved functions. In the example we've truncated the series at the fourth term and are ignoring terms of and higher.
In the example polynomial above the actual numbers are just made up numbers, as an example, and have no special significance - in case you are wondering why those particular numbers were used.
It turns out that truncating a series is in a certain sense not the best possible way to approximate the original function's actual values with a polynomial of degree three. A more complex but better way uses Chebyshev polynomials.
If instead we can express the function as:
where are constants then if we truncate this series at the fourth term, i.e T3, we again have a polynomial with no terms or higher. After we've expanded it out and collected the coefficients together, the coefficients will not generally be the same as when just truncating the series. It may sound crazy, so a worked example can show this more clearly.
Worked Example: Expansion of (2-x)-1
A quick check... does this formula look reasonable?
Put and we get .
Put and we get
.
The formula looks reasonable.
If we go to just four terms the error at is .
Now expressing this in Chebyshev polynomials:
Expanding this out we get:
to-do: add graph showing worst error is better. Unfortunately it isn't with the current calculation.
Hmm... still looks like I need to recompute the Chebyshev coefficients.
Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials Tn, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm.
Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials Tn, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm.
That cos(nx) is an nth-degree polynomial in cos(x) can be seen by observing that cos(nx) is the real part of one side of de Moivre's formula, and the real part of the other side is a polynomial in cos(x) and sin(x), in which all powers of sin(x) are even and thus replaceable via the identity cos2(x) + sin2(x) = 1.
This identity is extremely useful in conjunction with the recursive generating formula inasmuch as it enables one to calculate the cosine of any integral multiple of an angle solely in terms of the cosine of the base angle. Evaluating the first two Chebyshev polynomials:
and:
one can straightforwardly determine that:
and so forth. To trivially check whether the results seem reasonable, sum the coefficients on both sides of the equals sign (that is, setting equal to zero, for which the cosine is unity), and one sees that 1 = 2 − 1 in the former expression and 1 = 4 − 3 in the latter.
Two immediate corollaries are the composition identity (or the "nesting property")
and the expression of complex exponentiation in terms of Chebyshev polynomials: given z = a + bi,
Equivalently, the two sequences can also be defined from a pair of mutual recurrence equations:
These can be derived from the trigonometric formulae; for example, if , then
Note that both these equations and the trigonometric equations take a simpler form if we, like some works, follow the alternate convention of denoting our Un (the polynomial of degree n) with Un+1 instead.
A Chebyshev polynomial of either kind with degree n has n different simple roots, called Chebyshev roots, in the interval [−1,1]. The roots are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that
one can easily prove that the roots of Tn are
Similarly, the roots of Un are
One unique property of the Chebyshev polynomials of the first kind is that on the interval −1 ≤ x ≤ 1 all of the extrema have values that are either −1 or 1. Thus these polynomials have only two finite critical values, the defining property of Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:
The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it's easy to show that:
The last two formulas can be numerically troublesome due to the division by zero (0/0 indeterminate form, specifically) at x = 1 and x = −1. It can be shown that:
which, if evaluated as shown above, poses a problem because it is indeterminate at x = ±1. Since the function is a polynomial, (all of) the derivatives must exist for all real numbers, so the taking to limit on the expression above should yield the desired value:
where only is considered for now. Factoring the denominator:
Since the limit as a whole must exist, the limit of the numerator and denominator must independently exist, and
The denominator (still) limits to zero, which implies that the numerator must be limiting to zero, i.e. which will be useful later on. Since the numerator and denominator are both limiting to zero, L'Hôpital's rule applies:
The proof for is similar, with the fact that being important.
Indeed, the following, more general formula holds:
This latter result is of great use in the numerical solution of eigenvalue problems.
Concerning integration, the first derivative of the Tn implies that
and the recurrence relation for the first kind polynomials involving derivatives establishes that
For every nonnegative integer n, Tn(x) and Un(x) are both polynomials of degree n. They are even or odd functions of x as n is even or odd, so when written as polynomials of x, it only has even or odd degree terms respectively. In fact,
and
The leading coefficient of Tn is 2n − 1 if 1 ≤ n, but 1 if 0 = n.
Tn are a special case of Lissajous curves with frequency ratio equal to n.
In the appropriate Sobolev space, the set of Chebyshev polynomials form a completebasis set, so that a function in the same space can, on −1 ≤ x ≤ 1 be expressed via the expansion:[3]
Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which (among other things) implies that the coefficients an can be determined easily through the application of an inner product. This sum is called a Chebyshev series or a Chebyshev expansion.
Since a Chebyshev series is related to a Fourier cosine series through a change of variables, all of the theorems, identities, etc. that apply to Fourier series have a Chebyshev counterpart.[3] These attributes include:
The Chebyshev polynomials form a complete orthogonal system.
The Chebyshev series converges to ƒ(x) if the function is piecewisesmooth and continuous. The smoothness requirement can be relaxed in most cases — as long as there are a finite number of discontinuities in ƒ(x) and its derivatives.
At a discontinuity, the series will converge to the average of the right and left limits.
The abundance of the theorems and identities inherited from Fourier series make the Chebyshev polynomials important tools in numeric analysis; for example they are the most popular general purpose basis functions used in the spectral method,[3] often in favor of trigonometric series due to generally faster convergence for continuous functions (Gibbs' phenomenon is still a problem).
As an interpolant, the N coefficients of the (N − 1)th partial sum are usually obtained on the Chebyshev–Gauss–Lobatto[4] points (or Lobatto grid), which results in minimum error and avoids Runge's phenomenon associated with a uniform grid. This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:
The 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.
↑Chebyshev polynomials were first presented in: P. L. Chebyshev (1854) "Théorie des mécanismes connus sous le nom de parallélogrammes," Mémoires des Savants étrangers présentés à l’Académie de Saint-Pétersbourg, vol. 7, pages 539-586.