Gruber P. Convex and Discrete Geometry

498 Geometry of Numbers
is defined to be
E(J, F, N, W ) = sup
f ∈F
The minimum error then is
�� �
��
E(J, F, n) = inf
N⊆J,#N=n
W ⊆R,#W =n
J
f (x) dx − � ��
�
f (pi)wi � .
� E(F, N, W ) � .
Precise solutions of the problems to determine, for all n, the minimum error and to
describe the corresponding sets of nodes and weights are out of reach. Upper estimates
and asymptotic formulae as n →∞of the minimum error for several classes
F of functions have been given by Koksma and Hlawka, see Hlawka [515], by
Sobolev [946] and his school and by the Dnepropetrovsk school of numerical analysis,
see Chernaya [206, 207].
As a consequence of Zador’s theorem 33.2, we obtain the following result of
Chernaya [207]:
Corollary 33.4. Let 0 0, depending
only on α and d, such that the following statement holds: Let J ⊆ E d be a compact
Jordan measurable body and consider the following class H α of Hölder continuous
real functions on J:
Then
H α = � f : J → R :|f (x) − f (y)| ≤�x − y� α for x, y ∈ J � .
E(J, H α , n) ∼ δ
V (J)α+d d
n α d
i
as n →∞.
Proof. Taking into account Zador’s theorem, it is sufficient to prove that
(1) E(J, H α ⎧
⎨�
, n) = inf min
N⊆J ⎩ p∈N
#N=n
{�x − p�α ⎫
⎬
}dx
⎭ .
To see this, we proceed as follows
First, the following will be shown:
J
(2) Let N ={p1,...,pn} ⊆J and let h : J → R be defined by
h(x) = min
p∈N {�x − p�α }. Then h ∈ H α .
Let x, y ∈ J. By exchanging x and y, if necessary, we may assume that h(x) ≥ h(y).
Then
0 ≤ h(x) − h(y) = min
p∈N {�x − p�α }−min
q∈N {�y − q�α }
=�x − p� α −�y − q� α for suitable p, q ∈ N
≤�x − q� α −�y − q� α
=�x − y + y − q� α −�y − q� α ≤ (�x − y�+�y − q�) α −�y − q� α
≤�x − y� α +�y − q� α −�y − q� α =�x − y� α ,