Numerical Methods and Optimization

Polynomial Interpolation 123

1-1 0

x

y

1

FIGURE 5.2: Polynomial approximations to y = f(x)=

1

1+10x

2

over [−1, 1]

(shown with a solid line) based on 11 equally spaced nodes (dotted) and based

on 11 Chebyshev nodes (dashed).

4. T

n

(x) is an even function for n =2k, and an odd function for n =2k +1,

that is,

T

2k

(−x)=T

2k

(x)andT

2k+1

(−x)=−T

2k+1

(x),k=0, 1,....

It appears, that the normalized Chebyshev polynomial of degree n +1,

T

n+1

(x)/2

n

has a property useful for polynomial approximation: it has the

smallest deviation from zero over [−1, 1] among all normalized polynomials

of degree n + 1. This is expressed by the following theorem, which we state

without a proof.