Chebyshev Polynomial of the First Kind

A set of Orthogonal Polynomials defined as the solutions to the Chebyshev Differential Equation and denoted $T_n(x)$. They are used as an approximation to a Least Squares Fit, and are a special case of the Ultraspherical Polynomial with $\alpha=0$. The Chebyshev polynomials of the first kind $T_n(x)$ are illustrated above for $x\in[0,1]$ and $n=1$, 2, ..., 5.

The Chebyshev polynomials of the first kind can be obtained from the generating functions

$$g_1(t,x)\equiv {1-t^2\over 1-2xt+t^2} = T_0(x)+2\sum_{n=1}^\infty T_n(x)t^n$$ (1)

and

$$g_2(t,x)\equiv {1-xt\over 1-2xt+t^2} = \sum_{n=0}^\infty T_n(x)t^n$$ (2)

for $\vert x\vert \leq 1$ and $\vert t\vert < 1$ (Beeler et al. 1972, Item 15). (A closely related Generating Function is the basis for the definition of Chebyshev Polynomial of the Second Kind.) They are normalized such that $T_n(1)=1$. They can also be written

$$T_n(x)={n\over 2} \sum_{r=0}^{\left\lfloor{n/2}\right\rfloor } {(-1)^r\over n-r} {n-r\choose r} (2x)^{n-2r},$$ (3)

or in terms of a Determinant

$$T_n=\left\vert\matrix{ x & 1 & 0 & 0 & \cdots & 0 & 0\cr 1 &... ...ts & \vdots\cr 0 & 0 & 0 & 0 & \cdots & 1 & 2x\cr}\right\vert.$$ (4)

In closed form,

$$T_n(x) = \cos(n\cos^{-1}x)=\sum_{m=0}^{\left\lfloor{n/2}\right\rfloor } {n\choose 2m}x^{n-2m}(x^2-1)^m,$$ (5)

where ${n\choose k}$ is a Binomial Coefficient and $\left\lfloor{x}\right\rfloor$ is the Floor Function. Therefore, zeros occur when

$$x=\cos\left[{\pi (k-{\textstyle{1\over 2}})\over n}\right]$$ (6)

for $k=1$, 2, ..., $n$. Extrema occur for

$$x=\cos\left({\pi k\over n}\right),$$ (7)

where $k=0,1,\ldots, n$. At maximum, $T_n(x)=1$, and at minimum, $T_n(x)=-1$. The Chebyshev Polynomials are Orthonormal with respect to the Weighting Function $(1-x^2)^{-1/2}$

$$\int_{-1}^1 {T_m(x)T_n(x)\,dx\over\sqrt{1-x^2}} = \cases{ {... ...\delta_{nm}&for $m\not=0$, $n\not=0$\cr \pi& for $m=n=0$,\cr}$$ (8)

where $\delta_{mn}$ is the Kronecker Delta. Chebyshev polynomials of the first kind satisfy the additional discrete identity

$$\sum_{k=1}^m T_i(x_k)T_j(x_k) = \cases{ {\textstyle{1\over ... ...delta_{ij} & for $i\not=0$, $j\not=0$\cr m & for $i=j=0$,\cr}$$ (9)

where $x_k$ for $k=1$, ..., $m$ are the $m$ zeros of $T_m(x)$. They also satisfy the Recurrence Relations

$$T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$$ (10)

$$T_{n+1}(x)=xT_n(x)-\sqrt{(1-x^2)\{1-[T_n(x)]^2\}}$$ (11)

for $n\geq 1$. They have a Complex integral representation

$$T_n(x) = {1\over 4\pi i} \int_\gamma{(1-z^2)z^{-n-1}\,dz\over 1-2xz+z^2}$$ (12)

and a Rodrigues representation

$$T_n(x) = {(-1)^n\sqrt{\pi}\,(1-x^2)^{1/2}\over 2n(n-{\textstyle{1\over 2}})!} {d^n\over dx^n}[(1-x^2)^{n-1/2}].$$ (13)

Using a Fast Fibonacci Transform with multiplication law

$$(A,B)(C,D)=(AD+BC+2xAC,BD-AC)$$ (14)

gives

$$(T_{n+1}(x),-T_n(x))=(T_1(x),-T_0(x))(1,0)^n.$$ (15)

Using Gram-Schmidt Orthonormalization in the range ($-1$,1) with Weighting Function $(1-x^2)^{(-1/2)}$ gives

$$p_0(x) = 1$$ (16)

$$p_1(x) = \left[{x-{\int^1_{-1}x(1-x^2)^{-1/2}\,dx \over \int^1_{-1}(1-x^2)^{-1/2}\, dx}}\right]$$

$$= x-{[-(1-x^2)^{1/2}]^1_{-1}\over [\sin^{-1} x]^1_{-1}} = x$$ (17)

$$p_2(x) = \left[{x-{\int^1_{-1}x^3(1-x^2)^{-1/2}\,dx\over \int^1_{-1}x^2(1-x^2)^{-1/2}\,dx}}\right]x$$

$$\phantom{=} -\left[{\int^1_{-1}x^2(1-x^2)^{-1/2}\,dx\over \int^1_{-1}(1-x^2)^{-1/2}\,dx}\right]\cdot 1$$

$$= [x-0]x -{{\pi\over 2}\over \pi} = x^2-{\textstyle{1\over 2}},$$ (18)

etc. Normalizing such that $T_n(1)=1$ gives

$$\begin{eqnarray*} T_0(x) &=& 1\\ T_1(x) &=& x\\ T_2(x) &=& 2x^2-1\\ T_3(x... ... T_5(x) &=& 16x^5-20x^3+5x\\ T_6(x) &=& 32x^6-48x^4+18x^2-1. \end{eqnarray*}$$

The Chebyshev polynomial of the first kind is related to the Bessel Function of the First Kind $J_n(x)$ and Modified Bessel Function of the First Kind $I_n(x)$ by the relations

$$J_n(x)=i^n T_n\left({i{d\over dx}}\right)J_0(x)$$ (19)

$$I_n(x)=T_n\left({d\over dx}\right)I_0(x).$$ (20)

Letting $x\equiv\cos\theta$ allows the Chebyshev polynomials of the first kind to be written as

$$T_n(x)=\cos(n\theta)=\cos(n \cos^{-1} x).$$ (21)

The second linearly dependent solution to the transformed differential equation

$${d^2T_n\over d\theta^2}+n^2T_n=0$$ (22)

is then given by

$$V_n(x)=\sin(n\theta)=\sin(n\cos^{-1} x),$$ (23)

which can also be written

$$V_n(x)=\sqrt{1-x^2}\,U_{n-1}(x),$$ (24)

where $U_n$ is a Chebyshev Polynomial of the Second Kind. Note that $V_n(x)$ is therefore not a Polynomial.

The Polynomial

$$x^n-2^{1-n}T_n(x)$$ (25)

(of degree $n-2$) is the Polynomial of degree $<n$ which stays closest to $x^n$ in the interval $(-1,1)$. The maximum deviation is $2^{1-n}$ at the $n+1$ points where

$$x=\cos\left({k\pi\over n}\right),$$ (26)

for $k=0$, 1, ..., $n$ (Beeler et al. 1972, Item 15).

See also Chebyshev Approximation Formula, Chebyshev Polynomial of the Second Kind

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Orthogonal Polynomials.'' Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.

Arfken, G. ``Chebyshev (Tschebyscheff) Polynomials'' and ``Chebyshev Polynomials--Numerical Applications.'' §13.3 and 13.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 731-748, 1985.

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Iyanaga, S. and Kawada, Y. (Eds.). ``Cebysev (Tschebyscheff) Polynomials.'' Appendix A, Table 20.II in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1478-1479, 1980.

Rivlin, T. J. Chebyshev Polynomials. New York: Wiley, 1990.

Spanier, J. and Oldham, K. B. ``The Chebyshev Polynomials $T_n(x)$ and $U_n(x)$.'' Ch. 22 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 193-207, 1987.