By Richard V. Kadison
ISBN-10: 0821812076
ISBN-13: 9780821812075
Read Online or Download A Representation Theory for Commutative Topological Algebra PDF
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Extra resources for A Representation Theory for Commutative Topological Algebra
Sample text
In Section 2, we prove that these converge to f uniformly on [0, 1], and for twice continuously differentiable functions, obtain also the rate at which this convergence takes place. In Section 3, we prove an elegant theorem due to Korovkin and discuss some of its applications. 1. Bernstein Polynomials Definition 1. Let f: [O, 1] -+Rand n be a non-negative integer. We define the Bernstein polynomial of f of degree n by the formula Bn(/,z) := ~/ (~) (~)x•(l -z)"-•. (1) Proposition 2. Let Pn,a:(z) := (~)z•(1 - z)"-t ; " Tn,r(z) := L(k -nztPn,t(z).
Thus, II/ - (p* + ~q)lloo $ E - max (µ~, ~) < E. As explained before, this contradiction proves 'that if p• is a best approximation to / from IIn, then there exists a set of at least n + 2 alternation points. We now prove the converse statement. Thus, we assume that {x1} is an alternation set of n + 2 points. If p• is not a best approximation, then ti/ p*lloo =:A> E. 5, there exists a q E IIn such that II/ - ql1 00 = E. Then p*(x) - q(x) = /(x) - q(x) - (/(x) - p*(x)) alternates in sign n + 2 times at the x 1's.
6r1nf(o)(;)xr. Deduce from this that if J E IIm, then Bn(J;:z:) E IIm for all n. 1-3. 3 and Exercise 1 show that if then uniformly on [O, 1] for each k f E C(m)([0,1],R), = 0, 1, ... , m. 1-4. (Shoenberg)' HJ is convex in [0, 1], then for n = 2, 3, ... , Bn-1(f;x) ~ Bn(f;x),O < x < 1. Moreover, if f E C([O, 1], R), then strict inequality holds in the above inequality unless f is linear in each of the intervals lt~, n! 1],j = 1, 2, ... , n -1. In this case, we have Bn_ 1 (j;:z:) = Bn(f;x): 1-5. Let S = {z = (x1,x2,, ...