Chebyshev polynomials

There are two types of Chebyshev polynomials, of the first kind and of the second kind.

Chebyshev polynomials of the first kind are denoted T_n(x) and can be generated by expanding the expression:

                 -1
T (x) = cos(n cos  (x))
 n

This might not look like a polynomial at first glance, but through the use of trigonometric identities it may easily be seen to be:

       ⌊n/2⌋        n-2m   2   m
T (x) =  ∑    C   x     (x -1)
 n      m=0  n 2m

where _nC_r is a binomial coefficient, and ⌊n⌋ is the floor function.

These polynomials also are a special case of the Jacobi polynomials P^{(α, β)}_n where α and β are both -1/2. The Chebyshev T polynomials are also special cases of the hypergeometric function: T_n(x) = ₂F₁(-n, n; 1/2; (1-x)/2).

Chebyshev polynomials of the first kind are orthogonal over the interval -1<x<1with respect to the weight function (1-x²)^-1/2, i.e., the integral:

                      2 -1/2  
 1 T (x) T (x)  (1 - x )     dx
∫    m     n
-1

is equal to 1/2πδ_nm for m and n not zero, and π if m and n are identically zero (δ_mn denotes the Kronecker delta). The Chebyshev polynomials of the first kind also obey a discrete orthogonality relation:

 m
 ∑  T (x ) T (x )
k=1  i   k  j  k

is equal to 1/2mδ_ij for i and j not zero, and equal to m if i and j are both zero.

As with most orthogonal polynomials, the Chebyshev polynomials of the first kind also possess a three-term recurrence relationship:

T   (x) = 2xT (x) - T   (x)
 n+1         n       n-1

They also have a Rodrigues formula:

            n       2 1/2   n
        (-1) (π - πx )     d      2 n-1/2
T (x) = ----------------- ---((1-x )     )
 n         2n(n-1/2)!       n
                          dx

It also is one of the two linearly independent solutions to the second-order ordinary differential equation:

        2
    2  d y     dy    2
(1-x ) --- - x -- + n y = 0
         2     dx
       dx

(the other solution is the Chebyshev polynomial of the second kind, see below).

The orthogonality relations described and their relationship with the trigonometric functions make the Chebyshev polynomials of the first kind very important in the numerical approximation of functions, and numerical quadrature, as they provide a very fast and efficient way of computing high-quality approximations. They are also used in the construction of transfer functions for filters.

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The Chebyshev polynomials of the second kind are denoted U_n(x) are the other set of linearly independent solutions to the differential equation above. They also possess a Rodrigues formula:

                n      1/2      n
            (-1) (n+1)π        d        2 n+1/2
U (x) = ---------------------- --- ((1-x )     )
 n        n+1            2 1/2   n
         2   (n+1/2)!(1-x )    dx

They are also given by the summation:

       (n/2)         n-2r
U (x) =  ∑     C (2x)
 n      r=0 n-r r

The Chebyshev polynomials of the second kind are also special cases of the Jacobi polynomials P^(α,β)_n(x) where α and β are both 1/2. They are also special cases of the hypergeometric function, as U_n(x) = ₂F₁(-n, n+2; 3/2; (1-x)/2)

Chebyshev polynomials of the first kind are used in the design of Chebyshev Type I lowpass filters that are specified by maximum ripple within the passband.

The recurrence relationship for the n^th-order type I Chebyshev polynomial is

     T0(x) = 1
     T1(x) = x
     Tn(x) = 2⋅x⋅Tn-1(x) - Tn-2(x)

The first few polynomials are:

   ORDER     POLYNOMIAL
   -----     -----------------------
     0       T0(x) = 1
     1       T1(x) = x
     2       T2(x) = 2⋅x2 - 1
     3       T3(x) = 4⋅x3 - 3⋅x
     4       T4(x) = 8⋅x4 - 8⋅x2 + 1
     5       T5(x) = 16⋅x5 - 20⋅x3 + 5⋅x
     6       T6(x) = 32⋅x6 - 48⋅x4 + 18⋅x2 - 1
     7       T7(x) = 64⋅x7 - 112⋅x5 + 52⋅x3 - 7⋅x
     8       T8(x) = 128⋅x8 - 256⋅x6 + 152⋅x4 - 32⋅x2 + 1

Chebyshev polynomials of the second kind are used in the design of Chebyshev Type II lowpass filters that are specified by maximum ripple above the cutoff frequency.

The recurrence relationship for the n^th-order type II Chebyshev polynomial is

     U0(x) = 1
     U1(x) = 2x
     Un(x) = 2⋅x⋅Un-1(x) - Un-2(x)

The first few polynomials are:

   ORDER     POLYNOMIAL
   -----     -----------------------
     0       U0(x) = 1
     1       U1(x) = 2x
     2       U2(x) = 4⋅x2 - 1
     3       U3(x) = 8⋅x3 - 4⋅x
     4       U4(x) = 16⋅x4 - 12⋅x2 + 1
     5       U5(x) = 32⋅x5 - 32⋅x3 + 6⋅x
     6       U6(x) = 64⋅x6 - 80⋅x4 + 24⋅x2 - 1
     7       U7(x) = 128⋅x7 - 192⋅x5 + 80⋅x3 - 8⋅x
     8       U8(x) = 256⋅x8 - 448⋅x6 + 240⋅x4 - 40⋅x2 + 1

REFERENCES

1. George Arfken, Mathematical Methods for Physicists, 2nd Ed., Academic Press, (c)1970, p.626