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
Property 2. Leading Coefficient The coefficient of a in TN(a)is 2N-I whenN21
Property 2. Leading Coefficient
Property 3. Symmetry When N= 2M, T2M(a) is an even function, that is 12(-)=T2M(x) 1.77 When N= 2M +1, T2M+1(a)is an odd function, that is, T2M+1(-x)=TM+1(x)
Property 3. Symmetry
Property 4. Trigonometric 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()has n distinct zeros that lie in the interval [-1, 1(see Figure 4.15) 2k+1)丌 k= coS( 2M ) for k=0, 1,., M 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 81
Property 6. Extreme Values
