Best approximation of polynomial $f(x)$ by degree-one reduction

Question

Given a polynomial of $n+1$-th. order $f(x)$ on $-1 < x < 1$, I want to find the polynomial of nth. order $p_n(x)$ that minimizes the maximum error. This is just minimax approximation, except that $f(x)$ is a one degree higher polynomial. If $f(x) = x^{n+1}$, then $p_n(x) = x^{n+1} -\frac{T_{n+1}(x)}{2^n}$, where $T_n(x)$ are the Chebyshev polynomials.

My question is : If $f(x)$ is a general polynomial of degree $n+1$, what is $p_n(x)$ ?

The OP also asked about a more constrained problem in which the best max-norm approximation with matching endpoints is required. — hardmath, Jul 14 '13 at 01:45

score 2 · Answer 1 · answered Jul 07 '13 at 19:15

2

Hint: in general case, assuming that $f_{n+1}(x)=x^{n+1}+g_n(x)$ take $p_n(x)=g_n(x)+$ whatever you had in the previous case

answered Jul 07 '13 at 19:15

leshik

4,850

Best approximation of polynomial $f(x)$ by degree-one reduction

1 Answers1