CHAPTER 1 The result We consider the problem of characterization of best polynomial approximation on polytopes in Rd. To have a basis for discussion, first we briefly review the one-dimensional case. Let f be a continuous function on [−1, 1]. With ϕ(x) = 1 x2 and r = 1, 2,... let (1.1) ωr ϕ (f, δ) = sup 0h≤δ, x∈[−1,1] Δr hϕ(x) f(x) [−1,1] be its so called ϕ-modulus of smoothness of order r, where (1.2) Δr h f(x) = r k=0 (−1)k r k f x + ( r 2 −k)h is the r-th symmetric difference, and ·S denotes the supremum norm on a set S. In (1.1) it is agreed that Δr h f(x) = 0 if [x r 2 h, x + r 2 h] [−1, 1]. Let En(f)[−1,1] = inf pn f pn [−1,1] be the error of best approximation of f by polynomials pn of degree at most n. Then (see [12, Theorem 7.2.1]) for n r (1.3) En(f)[−1,1] Mωr ϕ f, 1 n and (see [12, Theorem 7.2.4]) (1.4) ωϕ r f, 1 n M nr n k=0 (k + 1)r−1Ek(f)[−1,1], n = 1, 2,..., where M depends only on r. (1.3)–(1.4) constitute what is usually called a characterization of the rate of best polynomial approximation in terms of moduli of smoothness, e.g. they give En(f)[−1,1] = O(n−α) ⇐⇒ ωr ϕ (f, δ) = O(δα) for α r. This is precisely what we want to do for multidimensional polynomial approximation in Rd. (1.3) is usually called the direct, or Jackson-type, while (1.4) is the converse, or Stechkin-type estimate. This latter (1.4) is a weak converse to (1.3), but that is natural, since En(f) can tend to zero arbitrarily fast, but ωϕ(f, r 1/n) c/nr unless f is a polynomial of degree at most r 1. In Rd we call a closed set K Rd a convex polytope if it is the convex hull of finitely many points. K is d-dimensional if it has an inner point, which we shall always assume. The analogue of the ϕ-modulus of smoothness on K was defined in [12, Chapter 12], and to recall its definition we need to consider the function along lines in different directions. A direction e in Rd is just a unit vector e Rd. Clearly, 3
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