TABLE OF CONTENTS In this section we derive lower bounds for the functionals ? nm and ? N, defined in Section A.2 , for calculating integrals (A.1) by the cubature formulas
| (A.11) |  |
and
| (A.12) |  |
on H lder and Sobolev classes.
Let ? = 
( D) or ? = 
( D), and calculate integral (A.1) by formula (A.11) with ? 1 = ? 2 = 0. Then the inequality
where 
, 
and
| (A.13) |  |
is valid.
Let ? = H ??( D) or 
, and calculate integral (A.1) by formula (A.11) with n 1 = n 2 = n and ? 1 = ? 2 = 0. Then the inequality
is valid.
Proof of Theorem A.5. Denote by ?( s 1, s 2) a nonnegative function belonging to the class 
(1) and vanishing at the nodes 
, 1 ? k 1 ? n 1, 1 ? k 2 ? n 2.
One has:
| (A.14) |  |
From Lemma A.5 and Theorem A.3 one concludes that the following inequality
holds for arbitrary weights 
and nodes 
and
Theorem A.5 is proved.
Let 
or 
, i = 1, 2, 3, and calculate the integral K f by cubature formula (A. 12). Then
where 
,
