By Carothers N.L.
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Extra resources for A short course on approximation theory (Math682)
Best Approximation 54 Even for relatively simple functions, the problem of actually nding the polynomial of best approximation is genuinely di cult (even computationally). We end this section by stating two important problems that Chebyshev was able to solve. Problem Find the polynomial pn;1 2 Pn;1, of degree at most n ; 1, that best approximates f (x) = xn on the interval ;1 1 ]. ) Since pn;1 is to minimize max jxn ; pn;1 (x)j, our rst problem is equivalent to: jxj 1 Problem Find the monic polynomial of degree n which deviates least from 0 on ;1 1 ].
Step 2. Given f 2 C 2 , there is a trig polynomial T such that 2f (x) sin2 x T (x). Each of the functions f (x) + f (;x) and f (x) ; f (;x)] sin x is even. Thus, we may choose even trig polynomials T1 and T2 such that f (x) + f (;x) T1(x) and f (x) ; f (;x)] sin x T2(x): Multiplying the rst expression by sin2 x, the second by sin x, and adding, we get 2f (x) sin2 x T1 (x) sin2 x + T2(x) sin x T3(x) where T3 (x) is still a trig polynomial, and where \ " now means \within 2"" (since j sin x j 1).
Also, it's easy to see that ;B (f ) (0) = f (0) and ;B (f ) (1) = f (1). In general, ;B (f ) (x) is an average of n n n the numbers f (k=n), k = 0 : : : n. Bernstein's theorem states that the sequence Bn(f ) converges uniformly to f for each f 2 C 0 1 ] the proof is rather simple once we have a few facts about the Bernstein polynomials at our disposal. For later reference, let's write f0 (x) = 1 f1(x) = x and f2(x) = x2 : Among other things, the following exercise establishes Bernstein's theorem for these three polynomials.