Preface
All four kinds of Chebyshev polynomials are orthogonal on the interval [−1, 1]
\[
\left\langle C_n (x) , C_m (x) \right\rangle = \int_{-1}^{+1} C_n (x)\,C_m (x)\,w(x)\,{\text d}x = \delta_{n,m} \| C_n \|^2 ,
\]
with respect to the following weight functions w(x) and norms
\( \| C_n (x) \|^2 : \)
\[
\mbox{for } \quad T_n (x): \qquad w(x) = \frac{1}{\sqrt{1-x^2}} \quad \mbox{and} \quad \| T_n (x) \|^2 = \begin{cases} \pi , & \ \mbox{ when } \quad n=0,
\\
\pi /2 , & \ \mbox{ for } \quad n > 0 ; \end{cases}
\]
\[
\mbox{for } \quad U_n (x): \qquad w(x) = \sqrt{1-x^2} \quad \mbox{and} \quad \| U_n (x) \|^2 = \frac{\pi}{2} ;
\]
\[
\mbox{for } \quad V_n (x): \qquad w(x) = \sqrt{\frac{1 + x^2}{1- x^2}} \quad \mbox{and} \quad \| V_n (x) \|^2 = \pi ;
\]
\[
\mbox{for } \quad W_n (x): \qquad w(x) = \sqrt{\frac{1 - x^2}{1+ x^2}} \quad \mbox{and} \quad \| W_n (x) \|^2 = \pi .
\]
Orthogonality of Chebyshev polynomials
The Chebyshev polynomials satisfy orthogonalities with respect to various weight functions as in the following:
\begin{align*}
\int_{-1}^1 \left( 1 - x^2 \right)^{-1/2} T_n (x) T_m (x) \,{\text d}x &=
\frac{\pi}{{\cal E}_n} \, \delta_{n,m} ,
\\
\int_{-1}^1 \left( 1 - x^2 \right)^{1/2} U_n (x) U_m (x) \,{\text d}x &=
\frac{\pi}{2} \, \delta_{n,m} ,
\\
\int_{-1}^1 \left( \frac{1+x}{1-x} \right)^{1/2} V_n (x) V_m (x) \,{\text d}x
&= \pi \, \delta_{n,m} ,
\\
\int_{-1}^1 \left( \frac{1-x}{1+x} \right)^{1/2} W_n (x) W_m (x) \,{\text d}x
&= \pi \, \delta_{n,m} ,
\end{align*}
where
\[
\delta_{n,m} = \begin{cases} 0, & \ \mbox{if } n \ne m ,
\\
1 , & \ \mbox{if } n = m ; \end{cases} \qquad\quad
{\cal E}_n = \begin{cases} 1 , & \ \mbox{if } n =0, \\
2, & \ \mbox{if } n \ge 1. \end{cases}
\]
For many applications the range [0, 1] is more convenient to use than [−1, 1]. Thus,
we map the independent variable x onto 2x −1 and label the Chebyshev polynomials
by an additional star
\[
C_n^{\ast} (x) = C_n \left( 2x -1 \right) \qquad \mbox{with} \qquad C^{\ast} = \left\{ T, \ U, \ V, \ W \right\} .
\]
The shifted Chebyshev polynomials are orthogonal
\[
\left\langle C_n^{\ast} (x) , C_m^{\ast} (x) \right\rangle = \int_{0}^{+1} C_n^{\ast} (x)\,C_m^{\ast} (x)\,w(x)\,{\text d}x = \delta_{n,m} \| C_n^{\ast} \|^2 ,
\]
with respect to the following weight functions w(x) and norms
\( \| C_n^{\ast} (x) \|^2 : \)
\[
\mbox{for } \quad T_n^{\ast} (x): \qquad w(x) = \frac{1}{\sqrt{1-x^2}} \quad \mbox{and} \quad \| T_n^{\ast} (x) \|^2 = \begin{cases} \pi , & \ \mbox{ when } \quad n=0,
\\
\pi /2 , & \ \mbox{ for } \quad n > 0 ; \end{cases}
\]
\[
\mbox{for } \quad U_n^{\ast} (x): \qquad w(x) = \sqrt{1-x^2} \quad \mbox{and} \quad \| U_n^{\ast} (x) \|^2 = \frac{\pi}{8} ;
\]
\[
\mbox{for } \quad V_n^{\ast} (x): \qquad w(x) = \sqrt{\frac{1 + x^2}{1- x^2}} \quad \mbox{and} \quad \| V_n^{\ast} (x) \|^2 = \frac{\pi}{2} ;
\]
\[
\mbox{for } \quad W_n^{\ast} (x): \qquad w(x) = \sqrt{\frac{1 - x^2}{1+ x^2}} \quad \mbox{and} \quad \| W_n^{\ast} (x) \|^2 = \frac{\pi}{2} .
\]
More generally, Chebyshev polynomials as well as other orthogonal polynomials
can be transformed to any given range [𝑎, b] via
\[
s(x) = \frac{2x - a -b}{b-a} \qquad \mbox{with} \quad x \in [a,b] \quad \mbox{and} \quad s \in [-1,1] .
\]
For example,
\[
U_n^{[a,b]} (x) = U_n (s(x)) .
\]
- Clenshaw, C.W., Norton, H.J.: The solution of nonlinear ordinary differential equations in chebyshev series. The Computer Journal, 1963, {\bf 6}, Issue 1, 88–92; https://doi.org/10.1093
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