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
Property 2. Leading Coefficient The coefficient of in TN()is 2- when N21
Property 2. Leading Coefficient
Property 3. Symmetry When N=2M, T2M(a)is an even function, that is I2M(-x)=12M(x) When N=2M +1, T2M+1( )is an odd function, that is 2M+1(-x)=T2M+1(x) (1.78)
Property 3. Symmetry
Representation on[-1,1] TN(x)=cos( N arccos(x)for-1≤x≤1. 1.79
Property 4. Trigonometric Representation on [-1,1]
Property 5. Distinct Zeros in [-1, 1 TN(r) has n distinct zeros that lie in the interval [-1, 1(see Figure 4.15) (2k+1)丌 k=coS ) for k=0, 1,., M 8 2M These values are called the Chebyshev abscissas(nodes)
Property 5. Distinct Zeros in [-1,1]
Property 6. Extreme Values TN(x)≤1for-1≤x≤1 (1.81)
Property 6. Extreme Values
