Abstract
Full Text
MATHEMATICS
I. M. SOBOL'
AN EXACT ESTIMATE OF THE ERROR OF MULTIDIMENSIONAL QUADRATURE FORMULAS FOR FUNCTIONS OF THE CLASS (S_p)
(Presented by Academician M. V. Keldysh on February 2, 1960)
Introduction. The error of an arbitrary quadrature formula
[
\delta(f)=\int_K f(P)\,dP-\sum_{i=1}^{N} C_i f(P_i)
]
is a linear functional, which we shall regard as given on some linear normed space (H) of functions defined on the unit (d)-dimensional cube (K). The norm (|\delta|) of this functional depends on the nodes (P_1,\ldots,P_N) and the weights (C_1,\ldots,C_N). Generally speaking, any problem of choosing the best quadrature formula (under any conditions) for the whole class (H) reduces to the problem of the minimum of (|\delta|) (under the corresponding conditions). A survey of results on extremal problems in the theory of quadrature formulas is given in the book of S. M. Nikol'skii ((^2)). All these results concern only the one-dimensional case ((d=1)).
In the present work formulas with equal weights (C_1=\cdots=C_N=1/N) are studied. It has proved possible to compute the norm of the error (for any (d)) for classes (S_p) of functions of many variables ((^4)). The geometric meaning of the dependence of the norm on the nodes is clarified. The question of the best order of convergence of quadrature formulas for the classes (S_p) and for the classes (H_\alpha) embedded in them is considered. For the class (S_1) a two-dimensional extremal problem is solved.
1. Main theorem. Choose arbitrary points (P_1,\ldots,P_N) from (K). Denote the coordinates of (P_i) by (x_{i1},\ldots,x_{id}). Introduce the function (\varphi_q(P_1,\ldots,P_N)), which will play the main role below.
Definition*.
[
\varphi_q(P_1,\ldots,P_N)
=
\sup_{m}^{\prime}
\left{
\sum_j
\left|
\sum_{i=1}^{N}
\operatorname{sgn}\,[\chi_{k_1}(x_{i1})\cdots \chi_{k_d}(x_{id})]
\right|^q
\right}^{1/q}.
\tag{1}
]
The prime indicates that the case (m_1=\cdots=m_d=0) is excluded when the upper bound is taken.
Let us estimate the error**
[
\delta_N(f)=\int_K f(P)\,dP-\frac{1}{N}\sum_{i=1}^{N} f(P_i).
\tag{2}
]
* All notation connected with the Haar functions (\chi_k(x)) corresponds to article ((^4)). For brevity, instead of ((m_1,\ldots,m_d)), ((j_1,\ldots,j_d)), ((k_1,\ldots,k_d)) we shall write (m,j,k).
** It would be more correct to write (\delta(f; P_1,\ldots,P_N)).
Theorem. Whatever the points (P_1,\ldots,P_N) in (K), for the class of functions (S_p)
[
|\delta_N|=\frac{\varphi_q(P_1,\ldots,P_N)}{N},\qquad \frac1p+\frac1q=1.
\tag{3}
]
Outline of the proof. First, it is necessary to establish that
[
|\delta_N(f)|\leq |f|_p\varphi_q(P_1,\ldots,P_N)/N .
\tag{4}
]
Expand (f(P)) in the uniformly convergent Haar series
[
f(P)=\sum_m \sum_i c_k\chi_{k_1}(x_1)\cdots \chi_{k_d}(x_d)
]
and substitute this series into (2). Interchanging the order of summation with respect to (i) and with respect to ((m,j)), we estimate (|\delta_N(f)|), applying Hölder’s inequality to the sum over (j). We obtain the inequality
[
|\delta_N(f)|\leq \frac1N \sum_m'\prod_{s=1}^d
2^{\frac{\widetilde m_s-1}{2}}
\left{\sum_j |c_k|^p\right}^{\frac1p}
\left{\sum_j\left|\sum_{i=1}^N
\operatorname{sgn}[\chi_{k_1}(x_{i1})\cdots \chi_{k_d}(x_{id})]\right|^q\right}^{\frac1q},
]
from which (4) follows.
Second, it is necessary to construct a function (g(P)\in S_p) for which inequality (4) becomes an equality. Fix a value (\widetilde m=(\widetilde m_1,\ldots,\widetilde m_d)) at which the least upper bound in formula (1) is attained (the existence of such an (\widetilde m) follows from item 2c). Denote
[
A_{\widetilde m j}=\sum_{i=1}^N
\operatorname{sgn}[\chi_{\widetilde m_1 j_1}(x_{i1})\cdots
\chi_{\widetilde m_d j_d}(x_{id})].
]
It is easy to verify that the function possessing the required property is
[
g(P)=\sum_j \operatorname{sgn} A_{\widetilde m j}\,
|A_{\widetilde m j}|^{q-1}
\chi_{\widetilde m_1 j_1}(x_1)\cdots
\chi_{\widetilde m_d j_d}(x_d).
]
2. Geometric definition of (\varphi_q(P_1,\ldots,P_N)). We shall regard the net (P_1,\ldots,P_N) as fixed. Whatever the volume (V\subset K), by (S_N(V)) we shall denote the number of points of the net lying in (V).
a) Suppose that all (m_s>0). By fixing the system of indices (m=(m_1,\ldots,m_d)), we thereby fix a partition of (K) into parallelepipeds
(\Pi_k\equiv \Pi_{k_1\ldots k_d}=l_{k_1}\times\cdots\times l_{k_d}), where
(l_{k_s}=l_{m_s j_s}). The sum over (j=(j_1,\ldots,j_d)) is the sum over all parallelepipeds of the given partition.
Move the origin of coordinates to the center (P') of the parallelepiped (\Pi_k). Let the new coordinates be (\xi_s). The coordinate planes (\xi_s=0) divide (\Pi_k) into (2^d) parallelepipeds. The sum (in the set-theoretic sense) of those parallelepipeds in which
(\operatorname{sgn}(\xi_1\cdots \xi_d)=(-1)^d) will be called the positive part of (\Pi_k) and denoted by (V_k^+); the remaining parallelepipeds constitute the negative part (V_k^-). Obviously, the volumes of these parts are equal.
If (P_i\in V_k^+), then
[
\operatorname{sgn}[\chi_{k_1}(x_{i1})\cdots \chi_{k_d}(x_{id})]
=(-1)^d\operatorname{sgn}(\xi_{i1}\cdots \xi_{id})=1.
]
If (P_i\in V_k^-), then this expression is equal to (-1), while if (P_i\notin \Pi_k), it vanishes. Consequently,
[
\sum_{i=1}^N \operatorname{sgn}[\chi_{k_1}(x_{i1})\cdots \chi_{k_d}(x_{id})]
= S_N(V_k^+)-S_N(V_k^-).
]
b) We shall call the quantity
[
\left{
\sum_{j_1,\ldots,j_d}
\left|
\sum_{i=1}^{N}
\operatorname{sgn}\,[\chi_{k_1}(x_{i1})\cdots \chi_{k_d}(x_{id})]
\right|^q
\right}^{1/q}
\tag{5}
]
in the case when all (m_s>0), the (d)-dimensional discrepancy of the points (P_1,\ldots,P_N) with respect to the partition ((m_1,\ldots,m_d)) of the (d)-dimensional cube (K). Consider the partition ((0,m_2,\ldots,m_d)), where all (m_s), except the first, are greater than zero. It is easy to see that expression (5) in this case is equal to
[
\left{
\sum_{j_2,\ldots,j_d}
\left|
\sum_{i=1}^{N}
\operatorname{sgn}\,[\chi_{k_2}(x_{i2})\cdots \chi_{k_d}(x_{id})]
\right|^q
\right}^{1/q}
]
and is the ((d-1))-dimensional discrepancy of the projections of the points (P_1,\ldots,P_N) onto the face (x_1=0) of the cube (K) with respect to the partition ((m_2,\ldots,m_d)) of this ((d-1))-dimensional face.
In order to compute (\varphi_q(P_1,\ldots,P_N)), it is necessary to find the greatest (d)-dimensional discrepancy of the points (P_1,\ldots,P_N) in (K), as well as the greatest (s)-dimensional discrepancies of the projections of the points (P_1,\ldots,P_N) onto all (s)-dimensional coordinate “faces” of the cube (K), (1\le s\le d-1). The function (\varphi_q(P_1,\ldots,P_N)) is equal to the largest of all these discrepancies.
c) It is not difficult to verify that (\varphi_q(P_1,\ldots,P_N)) coincides with the function (\varphi_q(N)), geometrically constructed in (1). The theorem of item 1 generalizes and refines Theorem 4.1 from ((^1)).
From the geometric definition some properties of (\varphi_q) are readily derived. We note that the upper bound in formula (1) is in fact taken only over a finite number of partitions. The function (\varphi_q) is bounded: for any (P_1,\ldots,P_N) from (K),
[
N^{1/q}\le \varphi_q(P_1,\ldots,P_N)\le N .
\tag{6}
]
- On the best grids for (S_p). In order to choose the best grid for a given number of nodes (N), it is necessary to find (\inf \varphi_q(P_1,\ldots,P_N)) over all possible grids (P_1,\ldots,P_N). This has so far been accomplished only in two special cases: a) for (d=1), (\inf \varphi_q(P_1,\ldots,P_N)=N^{1/q}) (cf. (6)); an evenly spaced grid may be taken as extremal; b) for (d=2), (\inf \varphi_\infty(P_1,\ldots,P_N)=2); consequently, the two-dimensional grids constructed in ((^1)), for which (\varphi_q=2^{1/p}N^{1/q}), are extremal with respect to the class (S_1).
Such two-dimensional grids can be constructed by the formulas
[
x_{ij}=y_{ji}=\frac{1}{n}(i+p_j),\qquad 0\le i,j\le n-1.
]
Here the number of nodes is (N=4^k); (n=\sqrt N); ({p_j}) is an infinite sequence of numbers, (p_0=0); (p_n=1/2n) for (n=2^k); (p_{n+i}=p_n+p_i) for (1\le i\le n-1).*
- On the best order of convergence for (S_p). One may pose a simpler problem: to find a family of grids (containing grids with arbitrarily large numbers of nodes) that ensures the best order of decrease of the error as (N\to\infty). According to (6), the best order of growth of (\varphi_q) cannot be less than (N^{1/q}). Grids for which (\varphi_q\asymp N^{1/q}) have been constructed only for (d=1) and (d=2). The very fact of their existence for (d\ge3) has not been proved. However, with the aid of the results of ((^3)), one can indicate multidimensional grids (for any (d)) for which (\varphi_q\le C N^{(1/q)+\varepsilon}), with (\varepsilon>0) arbitrarily small.
We note that, for the study of the order of convergence, it is sufficient to study (\varphi_\infty)—the simplest among all (\varphi_q). Indeed, from (1) it is not difficult
* Note added in proof. An analogous grid was constructed in Roth’s paper ((^5)) without connection with the problem of computing integrals.
derive that
[
\varphi_q(P_1,\ldots,P_N)\leqslant N^{1/q}\varphi_\infty(P_1,\ldots,P_N).
\tag{7}
]
Therefore, from (\varphi_\infty\leqslant CN^\varepsilon) it follows that (\varphi_q\leqslant CN^{(1/q)+\varepsilon}) for all (q).
5. Estimate of the error for functions of the class (H_\alpha). In (4) the classes of functions (H_\alpha) ((0<\alpha\leqslant 1)) are defined; they are analogues of the one-dimensional classes (\operatorname{Lip}\alpha), and the embedding theorem is proved: if (\alpha p>1), then (H_\alpha\subset S_p).
Theorem. Whatever the points (P_1,\ldots,P_N) from (K) and the function (f(P)) from (H_\alpha) may be,
[
|\delta_N(f)|\leqslant B\,\frac{\varphi_\infty(P_1,\ldots,P_N)\ln^d N}{N^\alpha},
\tag{8}
]
where (B\to L(e/2\alpha+1d\ln 2)^d) as (N\to\infty).
Sketch of proof. From inequalities (4), (7), and the estimate for (|f|_p) given in the embedding theorem, we obtain an estimate for (|\delta_N(f)|) containing (p) as a parameter, with (0<1/p<\alpha). For large (N) this estimate has its minimum at (1/p=\alpha-d\ln^{-1}N+O(\ln^{-2}N)), which is equal to the right-hand side of (8).
Estimate (8) is not sharp. However, there is an example showing that (\ln^d N) cannot be replaced by (\ln^{d-1}N) in (8).
From this theorem and item 4 it follows that there exist quadrature formulas ensuring, for the class (H_\alpha), the order of convergence (1/N^{\alpha-\varepsilon}). It is not difficult to prove that the order of convergence for the class (H_\alpha) cannot be better than (1/N^\alpha) (even if formulas with weights are used).
I consider it my pleasant duty to express gratitude to A. N. Tikhonov for his attention to this work.
Received
26 I 1960
CITED LITERATURE
- I. M. Sobol, DAN, 114, No. 4, 706 (1957).
- S. M. Nikol’skii, Quadrature Formulas, Moscow, 1958.
- N. M. Korobov, DAN, 124, No. 6, 1207 (1959).
- I. M. Sobol, DAN, 132, No. 4 (1960).
- K. F. Roth, Mathematica, 1, No. 2, 73 (1954).