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Speci cally, given a positive integer N , de ne L(x) by the conditions L(k=N ) = f (k=N ) for k = 0 1 : : : N , and L(x) is linear for k=N x (k +1)=N show that kf ; Lk is small provided that N is su ciently large. The function L(x) can be written (uniquely) as a linear combination of the \angles" P 'k (x) = jx ; k=N j + x ; k=N and 'N (x) = 1 the equation L(x) = Nk=0 ck 'k (x) can P be solved since the system of equations L(k=N ) = Nk=0 ck 'k (k=N ), k = 0 : : : N , Polynomials . 35 can be solved (uniquely) for c0 : : : cN .

It wasn't until 1903 that full details were given by Kirchberger. Curiously, Kirchberger's proofs foreshadow very modern techniques such as convexity and separation arguments. The presentation we'll give owes much to Haar and to de la Vallee Poussin (both from around 1918). We begin with an easy observation: Lemma. Let f 2 C a b ] and let p = pn be a best approximation to f out of Pn. Then, there are at least two distinct points x1 , x2 2 a b ] such that f (x1 ) ; p(x1 ) = ;(f (x2 ) ; p(x2 )) = kf ; pk: That is, f ; p attains both of the values kf ; pk.

For all practical purposes, they are one and the same space. While Bernstein's proof of the Weierstrass theorem (below) will prove most useful for our purposes, there are many others two of these (in the case of C 0 1 ]) are sketched below. R 19. (Landau's proof): For each n = 1 2 : : : and 0 1, de ne In( ) = 1 (1 ; x2 )n dx. Show that In( )=In (0) ! 0 as n ! 1 for any > 0. Now, given f 2 C 0 1 ] with R f (0) = f (1) = 0, show that the polynomial Ln(x) = (2In(0));1 01 f (t)(1;(t;x)2 )n dt converges uniformly to f (x) on 0 1 ] as n !

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A short course on approximation theory (Math682) by Carothers N.L.

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