1.5 Chebyshev Polynomials(Optiona
1.5 Chebyshev Polynomials (Optional)
Table 4.11 Chebyshev Polynomials To(a) through T(a) T0(x)=1 12(x)=2x2-1 T3(x)=4x3-3x T4(x)=8x4-8x2+1 1(x)=16x5-20x3+5x T6(x)=32x6-48x4+18x2-1 17(x)=64x7-112x5+56
1.5.1 Properties of Chebyshev Polynoals
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)=. and use the recurrence relation Tk(a)=2.cTk-1(a)-Tk-2(a)for k=2, 3,., 1.76
Property 1. Recurrence relation
Proof. Introducing the substitution 8 arccos(a) changes this equation to Tn(B(a )=Tm(0)=cos(n0), where 8E[0, T) A recurrence relation is derived by noting that Tn+1(0)=cos(ne)cos(0)-sin(ne )sin() ant T-1(0)=cos(ne)cos(0)+sin(ne)sin(g) O Tn+1(6)=2c0s(n6)cos()-Tn-1(6) Returning to the variable a gives Tn+1(a)=2 Tn()-Tn-1(a), for each n21