Second Kind Chebyshev Polynomials
Definition
$$U_{n} (x) := {{1} \over {n+1} } T_{n+1} ’ (x) = {{\sin \left( ( n +1 ) \theta \right)} \over { \sin \theta }} $$ is called the second kind Chebyshev polynomial.
Basic Properties
Recursive Formula
- [0]: $$U_{n+1} (x) = 2x U_{n} (x) - U_{n-1} (X)$$
Orthogonal Set
- [1] Inner product of functions: Given the weight $w$ for $\displaystyle \left<f, g\right>:=\int_a^b f(x) g(x) w(x) dx$ as $\displaystyle w(x) := \sqrt{1 - x^2}$, then $\left\{ U_{0} , U_{1}, U_{2}, \cdots \right\}$ forms an orthogonal set.
Chebyshev Nodes
- [2]: The roots of $\displaystyle U_{n} (X)$ are as follows for $k=1, \cdots , n$. $$x_{k} = \cos \left( {{k} \over {n+1}} \pi \right)$$
Even and Odd Functions
- [3]: $$U_{n} (-x) = (-1)^{n} U_{n} (x)$$
- Normally, for $0 \le \theta \le \pi$, it is assumed that $\theta := \cos^{-1} x $.
See Also
- First kind Chebyshev polynomial
- Second kind Chebyshev polynomial
- Relationship between the first and second kind Chebyshev polynomials
- Chebyshev polynomials as solutions to the Chebyshev differential equation
Explanation
The second kind Chebyshev polynomial for $n = 0, \cdots , 3$ is represented as follows.
$$ \begin{align*} U_{0} (x) =& 1 \\ U_{1} (x) =& 2x \\ U_{2} (x) =& 4x^{2} - 1 \\ U_{3} (x) =& 8x^{3} - 4x \end{align*} $$
$T_{n} (X)$ is the first kind Chebyshev polynomial.
Showing that $\displaystyle {{1} \over {n+1} } T_{n+1} ’ (x) = {{\sin \left( ( n +1 ) \theta \right)} \over { \sin \theta }}$ can be done using the differentiation of inverse trigonometric functions as follows. $$ \begin{align*} \displaystyle U_{n} (x) =& {{1} \over {n+1} } \left[ \cos \left( ( n +1 ) \cos^{-1} x \right) \right]’ \\ &= {{n+1} \over {n+1} } {{ - 1} \over { \sqrt{ 1 - x^{2} } }} \left[ - \sin \left( ( n +1 ) \cos^{-1} x \right) \right] \\ =& {{\sin \left( ( n +1 ) \cos^{-1} x \right)} \over { \sqrt{ 1 - x^{2} } }} \\ =& {{ \sin \left( (n+1) \theta \right) } \over {\sin \theta }} \end{align*} $$ The second kind Chebyshev polynomial is very useful not only in numerical analysis but also in applied mathematics as a whole, boasting interesting properties along with the first kind Chebyshev polynomial.
Meanwhile, the second kind Chebyshev polynomial can also be defined in reverse using $U_{0} (x) = 1$, $U_{1} (x) = 2x$, and the recursive formula [0]. This is also true for the first kind Chebyshev polynomial, and the reason for naming the first and second kinds is considered to be due to $T_{1} (x) = 1 \cdot x$ and $U_{1} (x) = 2 \cdot x$.
Proof
[0]
By differentiating both sides of the recursive formula $T_{n+1} (x) = 2x T_{n} (x) - T_{n-1} (X)$ of the first kind Chebyshev polynomial $$ T_{n+1} ' (x) = 2 T_{n} (x) + 2x T_{n} ' (x) - T_{n-1} ' (x) $$ since $T_{n+1} ' (x) = ( n+1 ) U_{n} (x) $, $$ (n+1) U_{n} (x) = 2 T_{n} (x) + 2x n U_{n-1} (x) - (n-1) U_{n-2} (x) $$ and combining as $n$, $$ n \left[ U_{n} (x) - 2x U_{n-1} (x) + U_{n-2} (x) \right] = 2 T_{n} (x) + U_{n-2} (x) - U_{n} (x) $$
Relationship between the first and second kind Chebyshev polynomials:
- [1]: $$U_{n} (x) - U_{n-2} (x) = 2 T_{n} (X)$$
$$ n \left[ U_{n} (x) - 2x U_{n-1} (x) + U_{n-2} (x) \right] = 0 $$ Dividing both sides by $n$ and rearranging yields $$ U_{n+1} (x) = 2x U_{n} (x) - U_{n-1} (x) $$
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[1]
Since $dx = - \sin \theta d \theta = - \sqrt{1 - x^2} d \theta$ and $\sin \theta = \sqrt{1 - x^2}$, $$ \begin{align*} \displaystyle \left< U_{n}, U_{m} \right> =& \int_{-1}^{1} U_{n} (x) U_{m} (x) \sqrt{1 - x^2} d x \\ =& - \int_{\pi}^{0} {{ \sin \left( (n + 1 ) \theta \right) \sin \left( (m + 1 ) \theta \right) \sin^2 \theta } \over { \sin^2 \theta}} d \theta \\ =& \int_{0}^{\pi} \sin \left( (n + 1 ) \theta \right) \sin \left( (m + 1 ) \theta \right) d \theta \\ =& \begin{cases} \pi/2 &, n=m \\ 0 &, n \ne m \end{cases} \end{align*} $$ thus $\left\{ U_{0} , U_{1}, U_{2}, \cdots \right\}$ forms an orthogonal set.
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[2]
It is self-evident by definition.
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[3]
Case 1. $n=0,1$
$$ \begin{align*} U_{0} (-x) =& 1 = U_{0} (x) \\ U_{1} (-x) =& 2(-x) = -2x = - U_{1} (x) \end{align*} $$
Case 2. $n \ge 2$ is even
Since the degree of all terms that are not $0$ in $U_{n}(x)$ is even, $U_{n}(-x) = U_{n}(x)$
Case 3. $n \ge 2$ is odd
Since the degree of all terms that are not $0$ in $U_{n}(x)$ is odd, $U_{n}(-x) = - U_{n}(x)$
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Implementation
Below is the code for the Chebyshev polynomial written in R.
Since it returns the polynomial itself, it can be directly used for calculations. n
is the degree, and by setting kind
and turning the print
option to true, it will print the coefficients.
The printed coefficients are from the constant term to the higher-order terms in sequence, and since the second kind Chebyshev polynomial is $U_{3} (x) = 8x^{3} - 4x$, it is correct. The value of the function is also accurately calculated as $U_{3} (3) = 8 \cdot 3^{3} - 4 \cdot 3 = 216-12 = 204$.
Chebyshev<-function(n,kind=1,print=F)
{
p<-NA
if((round(n)-n)!=0 | n<0) {stop("Wrong Degree!!")} #degree must be nonnegative integer
if(!kind%in%(1:2)) {stop("Wrong Kind!!")} #kind must be 1 or 2
if(n==0)
{
if(print) {print(1)}
p<-function(x) {return(1)}
return(p)
}
if(n==1)
{
if(print) {print(c(0,kind))}
p<-function(x) {return(kind*x)}
return(p)
}
coef0<-c(1)
coef1<-c(0,kind)
for(i in 1:(n-1))
{
coef2<- ( c(0,2*coef1) - c(coef0,0,0) )
coef0<-coef1
coef1<-coef2
}
if(print) {print(coef2)}
p<-function(x) {return(sum(coef2*x^(0:n)))}
return(p)
}
p<-Chebyshev(1,2); p(2)
p<-Chebyshev(3,2,T); p(3)