1.5 Chebyshev polynomials(optional)
1.5 Chebyshev Polynomials (Optional)
Table 4.11 Chebyshev Polynomials To() through T7(a) 10(x)=1 1(x)=x 12(x)=2ax2-1 13(x)=4x3-3x T4(x)=8x4-8x2+1 I5(x)=16x5-20x3+5x (x)=32x6-48x4+18x2-1 7(x)=64x7-112x5+56x3-7x
1. 5. 1 Properties of Chebyshev polynomials
1.5.1 Properties of Chebyshev Polynomials
Property 1. Recurrence relation Chebyshev polynomials can be generated in the following way. Set To(a)=1 and Ti(a)=a and use the recurrence relation Tk(x)=2xh-1(x)-1k-2(x)fork=2,3, (1.76)
Property 1. Recurrence relation
Proof. Introducing the substitution 8 =arccos( changes this equation Tn(0(a))=T(0)=cos(ne), where 0E[ 0, a recurrence relation is derived by noting that Tn+1(6)=c8(76)cs6)-sin(n)sin(6) ane Tm-1(0)=cos(nl)cos(0)+sin(ne)sin(A Tn+1(6)=2c0s(n6)o(6-Tn-1(6). Returning to the variable g gives Tn+(a)=2 Tn()-Tn-1(), for each n2