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Encyclopedia > Chebyshev form

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are special polynomials. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and Chebyshev polynomials of the second kind which are denoted Un. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebyshef or Tschebyscheff.


The Chebyshev polynomials Tn or Un are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence.


Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides the best approximation to a continuous function under the maximum norm.


In the study differential equations they arise as the solution to the Chebyshev differential equation

for the polynomials of the first and second kind, respectively. These equation are special cases of the Sturm-Liouville differential equation.

Contents

Definition

The Chebyshev polynomials of the first kind are defined by the recurrence relation

One example of a generating function for this recurrence relation is

The Chebyshev polynomials of the second kind are defined by the recurrence relation

One example of a generating function for this recurrence relation is

Examples

This image shows the first few Chebyshev polynomials of the first kind in the domain _1¼<x<1¼, _1¼<y<1¼; the flat T0, and T1, T2, T3, T4 and T5.
This image shows the first few Chebyshev polynomials of the first kind in the domain _1¼<x<1¼, _1¼<y<1¼; the flat T0, and T1, T2, T3, T4 and T5.

The first few Chebyshev polynomials of the first kind are

This image shows the first few Chebyshev polynomials of the second kind in the domain _1¼<x<1¼, _1¼<y<1¼; the flat U0, and U1, U2, U3, U4 and U5. Although not visible in the image, Un(1)=n+1 and Un(_1)=(n+1)(_1)n.
This image shows the first few Chebyshev polynomials of the second kind in the domain _1¼<x<1¼, _1¼<y<1¼; the flat U0, and U1, U2, U3, U4 and U5. Although not visible in the image, Un(1)=n+1 and Un(_1)=(n+1)(_1)n.

The first few Chebyshev polynomials of the second kind are

Trigonometric definition

The Chebyshev polynomials of the first kind can be defined by the trigonometric identity

for n = 0, 1, 2, 3, .... . 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.


Written explicitly

Similarly, the polynomials of the second kind satisfy

Notes

The Chebyshev polynomials of the first and second kind are closely related by the following equations

Both the Tn and the Un form a sequence of orthogonal polynomials. The polynomials of the first kind are orthogonal with respect to the weight

on the interval [−1,1], i.e., we have

This is because (letting x = cos θ)

Similarly, the polynomials of the second kind are orthogonal with respect to the weight

(which, when normalized to form a probability measure, is the Wigner semicircle distribution). The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomials, which themselves are a special case of the Jacobi polynomials.


Polynomial in Chebyshev form

A polynomial of degree N in Chebyshev form is a polynomial p(x) of the form

where Tn is the nth Chebyshev polynomial.


Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.


Chebyshev roots

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 form one can easily prove that the roots of Tn are

Similarly, the roots of Un are



See also

References

  • M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Chapter 22. New York: Dover, 1972.



  Results from FactBites:
 
Chebyshev biography (3026 words)
Chebyshev always acknowledged the great influence Brashman had been on him while studying at university, and credited him as the main influence in directing his research interests, referring to their "precious personal talks".
Chebyshev continued to aim at international recognition with his second paper, written again in French, appearing in 1844 published by Crelle in his journal.
In the summer of 1846 Chebyshev was examined on his Master's thesis and in the same year published a paper based on that thesis, again in Crelle's journal.
Chebyshev (234 words)
In 1847 Chebyshev was appointed to the University of St Petersburg.
Chebyshev also came close to proving the prime number theorem, proving that if pi (n)log n)/n had a limit as n-> infinity then that limit is 1.
Chebyshev was also interested in mechanics and studied the problems involved in converting rotary motion into rectilinear motion by mechanical coupling.
  More results at FactBites »

 
 

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