The Chebyshev polynomials named after Pafnuty Chebyshev (Пафнутий Чебышёв), compose a polynomial sequence, and are 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.
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:
T0(x)=1
T1(x)=x
T2(x)=2x2−1
T3(x)=4x3−3x
T4(x)=8x4−8x2+1
T5(x)=16x5−20x3+5x
T6(x)=32x6−48x4+18x2−1
T7(x)=64x7−112x5+56x3−7x
T8(x)=128x8−256x6+160x4−32x2+1
T9(x)=256x9−576x7+432x5−120x3+9x
