Chebyshev Polynomials

The Chebyshev polynomials are two sequences of polynomials, T_n and U_n. The Chebyshev polynomials of the first kind, T_n, can be defined by the recurrence relation T_0(x)=1, T_1(x)=x, and T_{n+1}(x) = 2xT_n{x}-T_{n-1}(x). The Chebyshev polynomioals of the second kind, U_n(x), can be defined by U_0(x)=1, U_1(x)=2x, and U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x). Both T_n and U_n have degree n, and any polynomial of degree n may be uniquely written as a linear combination of the polynomials T_0, T_1, ..., T_n (similarly with U).

First Kind

ChebyshevT{<:Number}(coeffs::AbstractVector, var=:x)

julia> using Polynomials

julia> ChebyshevT([1, 0, 3, 4])
ChebyshevT(1⋅T_0(x) + 3⋅T_2(x) + 4⋅T_3(x))

julia> ChebyshevT([1, 2, 3, 0], :s)
ChebyshevT(1⋅T_0(s) + 2⋅T_1(s) + 3⋅T_2(s))

julia> one(ChebyshevT)
ChebyshevT(1.0⋅T_0(x))

The ChebyshevT type holds coefficients representing the polynomial a_0 T_0 + a_1 T_1 + ... + a_n T_n.

For example, the basis polynomial T_4 can be represented with ChebyshevT([0,0,0,0,1]).

Conversion

ChebyshevT can be converted to Polynomial and vice-versa.

julia> c = ChebyshevT([1, 0, 3, 4])
ChebyshevT(1⋅T_0(x) + 3⋅T_2(x) + 4⋅T_3(x))


julia> p = convert(Polynomial, c)
Polynomial(-2 - 12*x + 6*x^2 + 16*x^3)

julia> convert(ChebyshevT, p)
ChebyshevT(1.0⋅T_0(x) + 3.0⋅T_2(x) + 4.0⋅T_3(x))