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) ωϕ(f,
r
δ) = sup
0h≤δ, x∈[−1,1]
Δhϕ(x)f(x)
r
[−1,1]
be its so called ϕ-modulus of smoothness of order r, where
(1.2) Δhf(x)
r
=
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 Δhf(x)
r
= 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] Mωϕ
r
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)
⇐⇒ ωϕ(f,
r
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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