Second Kind Chebyshev Polynomials 📂Numerical Analysis

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 $\theta := \cos^{-1} x$ .

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.

$20181120\_130322.png$

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)