Chebyshev polynomials
The
Chebyshev polynomials are named after
Pafnuty Chebyshev (
Пафнутий Чебышёв) and compose a
polynomial sequence. This article refers to what are commonly known as
Chebyshev polynomials of the first kind, which are a solution to the Chebyshev
differential equation:
They can be defined by
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. The polynomial
Tn has exactly
n simple roots in [−1, 1] called
Chebyshev roots. The Chebyshev polynomials can be used in the area of
numerical approximation.
Alternatively they can be defined via the recurrence relation
-
-
These
polynomials are orthogonal with respect to the weight
on the interval [−1,1], i.e., we have
This is because (letting
x = cos θ)
The first few polynomials are:
 = 16x^5 - 20x^3 + 5x \\,.png)
One example of a
generating function for this
polynomial sequence is
See also
- Chebyshev nodes
- Legendre polynomials
- Hermite polynomials