CC BY
6
DOI: 10.17516/1997-1397-2020-13-4-398-413 YflK 519.644.7
On Error Estimates in Sp for Cubature Formulas Exact
for Haar Polynomials
Kirill A. Kirillov*
Siberian Federal University Krasnoyarsk, Russian Federation
Received 12.03.2020, received in revised form 8.05.2020, accepted 15.05.2020 Abstract. On the spaces Sp, an upper and lower estimates for the norm of the error functional cubature formulas possessing the Haar ¿-property are obtained for the n-dimensional case.
Keywords: Haar ¿-property, error estimates for cubature formulas, function spaces Sp.
Citation: K.A. Kirillov, On Error Estimates in Sp for Cubature Formulas Exact for Haar Polynomials,
J. Sib. Fed. Univ. Math. Phys., 2020, 13(4), 398-413. DOI: 10.17516/1997-1397-2020-13-4-398-413.
Introduction
The problem of constructing and analyzing cubature formulas that are exact for a given set of functions was earlier considered primarily as applied to the computation of integrals exact for algebraic and trigonometric polynomials. For example, the approximate integration formulas of algebraic accuracy can be found in [1,2]. The cubature formulas exact for trigonometric polynomials in particular were studied in [3-7].
The approximate integration formulas exact for the system of Haar functions can be found in the monograph [8]. The accuracy of approximate integration formulas for finite Haar sums was used in [8] to derive error estimates for these formulas.
A description of all minimal weighted quadrature formulas possessing the Haar ¿-property, i.e., formulas exact for Haar functions of groups with indices not exceeding a given number d, was given in [9]. The error estimates for quadrature formulas possessing the Haar d-property in the case of the weight function g(x) = 1 were obtained in [10]. In particular, in the mentioned paper
the upper estimate for the norm of the error functional \\SN|Lt was found for the quadrature
p
formulas having the Haar d-property:
\\SN\\s. < (2df p, p
and the lower estimate for the norm of the error functional \\SN\\s„ was obtained for the quadrature formulas exact for constants:
\\Sn\\s, > 2-pN-p. p
The problem of constructing cubature formulas possessing the Haar d-property, i.e., formulas exact for Haar polynomials of degree at most d, was solved in the two-dimensional case in
* kkirillow@yandex.ru https://orcid.org/0000-0002-3763-1303 © Siberian Federal University. All rights reserved
[11-15] under the condition that the weight function g(x1, x2) = 1. The error estimates for these cubature formulas was derived in [16]. In particular, in [16] the upper estimate for the norm of the error functional \\SN IL, was obtained for the mentioned cubature formulas:
\\<NIL < 21 (2d)-1. p
In the present paper the error estimates of cubature formulas with arbitrary positive coefficients at the nodes, similar to the estimates given above for the one- and two-dimensional cases, are derived in the n-dimensional case. As a result, we find the upper estimates for the error functional SN of the cubature formulas possessing the Haar d-property:
n- 1 / 7\ - 1 n- 1 / 7\ - 1
\SN [f]\ < 2— (2d) p \\f \L, \\SNIL, < 2~ (2d) p ,
and we obtain the lower estimate for the norm of the error functional \\SN\L, for the cubature formulas exact for any constant:
\\Sn\L > (2"+1 - n - 1) p N-p.
1. Basic definitions
In this paper, we use the original definition of the functions xm,j (x) introduced by A. Haar [17].
r 1 \ / 2m-1 — 1
The binary intervals of rank m are the intervals lmj1 = 0, ^m-I I, = ( —2m—i—' 1
j — 1 j \ ... o A • o nm-1
m = 2, 3,..., and lm,j = ^2m-1, 2m—ij , m = 3, 4,..., j = 2,..., 2m — 1. By a binary interval of the 1st rank we will consider the interval l1t1 = [0,1]. The binary segments of rank m are the
closed intervals lm,j =
j - 1
1; , , m =1, 2,...,j = 1,..., 2m-1.
2m— 1 7 2™-1 777./ 11
The left and right halves of lm,j (without its midpoint) are denoted by lm j and l, respectively. Obviously, l-, j = lm+1,2j-1, ,j = lm+1,2j. In [17], the Haar functions xm,j (x) are defined by:
2 ^, x e lm jj,
—2 ^ , x e lm j,
Xm,j (x)
_ (1)
0, x e [0,1] \ lmj,
{Xm,j(x — 0) + Xm,j(x + 0)}/2, x is an interior discontinuity point,
m =1, 2,..., j = 1,..., 2m-1.
Thus, the Haar system of functions is constructed in groups: the mth group contains 2m-1 functions {xm,j (x)}, where m = 1, 2,..., j = 1,..., 2m-1. The Haar system of functions includes the function x1(x) = 1 too, which is outside of any group.
In the one-dimensional case, the Haar polynomials of degree d are by definition the functions
d 2m-1
pd(x) = ao + a™Xmj(x),
m=1 j=1
where d = 1, 2,..., a0, al G R, m = 1,...,d, j = 1,..., 2m-1, and
2d-i
E(adj)}2 =0.
j=i
By the 0-degree Haar polynomials we will consider real constants.
In the n-dimensional case, the Haar polynomials of degree d are the functions
Pd(xi, ...,xn) = ao+
n 2mi-1 2ms-1
+ y ] y ] ^ ] ^ ] . . . y ] amii\... 'jr^s (i1, . . . , is)Xmi,ji (xii ) ...Xms,js (xis ),
s = 1 1<ii< . . . <is ^n mi + . . . +ms^.d ji=1 js = 1
where d =1, 2, . . . , a0, ami'.. . . 'm (i1,... ,is) G R, 1 ^ i1 < ... < is ^ n, m1 + ... + ms ^ d, s = 1,...,n, jk = 1,..., 2mk-1, k = 1, ...,s, and
n 2mi-ii 2ms-i 2
EE E (i1,...,is)} =0.
s=1 1^ii<...<is^n mi + ...+ms =d ji=1 js = 1
The same way as in the one-dimensional case, by 0-degree Haar polynomials we will consider real constants.
Consider the following cubature formula
C1 C1 N / \
I [f] = ... f(x1,...,xn) dx1 ...dxn «V Ckf (x{k),...,xik)) = Qn [f], (2)
Jo Jo k=1 V J
where (x1k\ ..., xk) G [0,1]n are the nodes, the coefficients C\ at the nodes are real, k = 1,...,N.
The cubature formula (2) is said to possess the Haar d-property (or just the d-property) if it is exact for any Haar polynomial P(x1,... ,xn) of degree at most d, i.e., QN[P] = I[P]. Such a formula with the least possible number of nodes is called a minimal cubature formula with the d-property.
We recall the definition of the linear normed space Sp in the n-dimensional case introduced by I.M.Sobol' [8].
Let p be a fixed number with 1 < p < The set of functions f (x1,..., xn) defined in the unit n-dimensional cube [0,1]n and representable as a Fourier-Haar series
f(xi, ...,Xn) = co+
2mi-
s -1
+ ■ ■ ■ ■ ■ ■ cni11',...',tns ■ ■ ■ , is)Xmi,ji (xii ) ■■■Xms ,js (xi s )
(3)
s = l l<ii<. . . <is^nm1=l ms = l j1 = l js = l
with real coefficients c0, 0^,1'.. '. '. 'jms (il ¡■■■,is) (1 ^ il < ■ ■ ■ < is < n, ml, ■ ■ ■ ,ms = 1, 2, ■ ■ ■. s = 1, ■ ■ ■ ,n, jk = 1, ■ ■ ■, 2mk _l, k = 1, ■ ■ ■ ,s) satisfying the conditions
Api1'...'is)(f ) =
s
"p
'2m1-1 2ms-1 ) p
E ■■■ £ E ■■■ E jj(il.....is)^" <Ai1'...'is, (4)
m1 = l ms = l (, j1 = l js = l )
2
(where Ai1..,is are real constants, 1 < i1 < ... < is < n, 1 < s < n) is called the class
Sp(A1, . . . , An, . . . , Ai1,...,is . . . , A1,..,n).
It was proved in [8] that the set of functions f (x1,... ,xn) belonging to all the classes SP(A1 ,...,An,... ,Ai1,...,is ...,A1,...,n) (with all possible A1,..., An,..., Ah>...,is,..., A^...^, while p being fixed) equipped with the norm
n
\\f\Lp = Y E ApPi1 -is)(f), (5)
s=1 <...<is ^n
forms a linear normed space, which is denoted by Sp. All the functions f (x1,... ,xn) that differ by constant terms are regarded as a single function.
The coefficients c0, ctims (i1,...,is) (1 < i1 < ... < is < n, m1,...,ms = 1, 2,..., s = 1,... ,n, jk = 1,..., 2mk-1, k = 1,..., s) in the representation of the function f (x1,..., xn) as a series (3) are called the Fourier-Haar coefficients of this function.
In [8] it was proved that the series (3) converges absolutely and uniformly.
2. Derivation of estimates for the norm of the error functional of cubature formulas in Sp
Let (2) be a cubature formula with the coefficients Ck at the nodes satisfying the inequalities Ck > 0, k = 1, 2,... ,N. We denote the error functional of the cubature formula (2) by SN [f] so that
r 1 r 1 N / \
Sn [f] = I [f] — Qn [f] = .. f(xu...,xn) dx1 ...dxn — V Ckf (x[k),...,xSk) , (6)
Jo Jo k=1 V 7
where the function f e Sp, p > 1. It was shown in [8] that any such function is continuous at all points which coordinates are not binary rational numbers. Hence the integral
1 1
f ... f f (x1,..., xn) dx1 ... dxn exists not only in the Lebesgue sense, but also in the Riemann
oo sense.
Let
2m1—1 2ms—1
E
j1 = 1 js = 1
r ■ N m1
^..m (q) = 2 2
...
N
E Ck Xm1 ¿^i^) . . . Xmsdsi^i^
k = 1
9 1 q
(7)
where q > 1, 1 ^ ¿1 < ... < is ^ n, m1,..., ms = 1, 2,..., s = 1,..., n.
Lemma 1. If the cubature formula (2) is exact for any constant and f e Sp, then for the absolute value of the error functional satisfies the inequality
œ œ
I'nf <E E E - E 2=v
s=1 1^i1<...<is^n t1 = 1 m s = 1 I
2 X
■2m1-
E - E jj(n,...,is)
- j1 = 1 js = 1
(8)
zm\';....m (q) .
Proof. The series (3) is substituted into (6). Since the series (3) converges uniformly and since the cubature formula (2) is exact for any constant, we have:
1
1
2
1
2
p
P
x
2mi-
[f] = -E E ••• E {jj&>■■■>*>)
s=1 ... <is^nm1 = l ms = 1 ji = l js = 1
N s
^E CkXmi,ji (x^) •••Xms,js (x^) >•
k=1 '
(9)
Since the series in (3) is absolutely convergent, it follows that the series in (9) also absolutely converges. Applying the triangle inequality to the expression on the right-hand side of (9), we obtain:
^mi —1 oms—1
n to to 2mi—1 2
I'n E E-EE-E
s=1 1^i1<...<is^nm1 = 1 ms = 1 j1 = 1 js = 1
N
^ E CkXm1,j1 (x^) • ••Xms,js
(10)
k=1
Now we apply the Holder inequality to the sums over jt,.. .,js on the right-hand side of (10). Taking into account (7), we obtain the inequality (8). □
It was shown in [9] that there exist Haar polynomials of one variable of degree m that satisfy the equality:
2 , ^ lm
Km,jfa} \ 2
m — 1
lm+1,j , £ lm+1,j \ lm+1,j 7
(11)
0, x £ [0,1] \ lm+1j,
where m = 1,2,... and j = 1,2,..., 2m. It was also proved in [9] that the functions Km:1(x),..., Km,2m (x) form a basis in the linear space of Haar polynomials of degree at most m.
The definition of the Haar functions (1) and relation (11) imply the following equalities:
Xm,j (Xi)=2 2 Km,2j-1 (Xi) - Km,2j (xi) Km,2j-1(xi) + Km,2j (xi) 2Km-1,j (xi),
i = 1,...,n, m =1, 2,..., j = 1,..., 2m-1. Let
1 :...:js )
(12) (13)
^-m1',... j}s (xi1 , • • • , xis ) Km1,j1 (xi1 ) • • • Kms,js (xis ),
is/ ^m1,j1V- i1>--- '^msjsK^ ish (14)
1 ^ i1 < • • • < is ^ n, m1, ..., ms = 1, 2,. .., s = 1,.. . ,n, jr = 1,.. •, 2mr-1, r = 1,. . • , s.
Lemma 2. For any ordered set (it,..., is), 1 ^ it < ... < is ^ n, 1 ^ s ^ n, and for any positive integer M there exists at least one ordered set (Mt,..., Ms) satisfying the inequality Mt + ... + Ms > M such that
Zmu/'m. (q)= sup ^ sm\';;::im)s (q).
m 1 +...+ms "^M
(15)
Proof. For a fixed positive integer M, we choose (int,... ,ms) in accordance with condition that the sum mt +... + ms is minimum among all ordered sets (mt,... ,ms) such that mt +... + +ms > M and each of the closed s-dimensional binary parallelepipeds lm 1 + t,j 1 x ... x lms+t,js contains at most one node of the cubature formula (2).
2
x
x
If the coordinates of the nodes of the cubature formula (2) xk e {2 mr (2jr — 0) : jr = = 1,..., 2mr-1}, k = 1,..., N, then we set mr = mr. Otherwise, we set fhr = 1 +max{mr e N : there exists x{rK) = 2-m (2j[K) — 1), 1 < j^) < 2m-1, 1 < K < N}, r =1,...,s.
Then, for all ordered sets (m1,... ,ms) such that m1 + ... + ms > mn 1 + ... + mn s the following three conditions are satisfied:
- the inequality m1 + ... + ms > M holds;
- each of the closed s-dimensional binary parallelepipeds lmi + 1,j1 x ... x lms+1,js contains at most one node of the cubature formula (2);
- the coordinates of every node of the cubature formula (2) differ from the points {2-mr (2jr — 1)} = supp {Kmr ,2jr-1 }n supp {Kmr ,2jr }, jr = 1,..., 2mr-1,
By virtue of (7), (12), we have:
r = 1, .. ., s.
y (¿1 ,...,is) mi ,...,ms
(k)
(q) = 2
2mi-1 2ms-1
-m 1 -...-m^ J
1 j1=1 js = 1
N
E Ck x
k=1
m 1,2j1-1 lxi1 ) Km 1,2j1
(k)
(k) (k) ,2js-1 < — Km s,2js <
(16)
1 ^ i1 < . .. < is ^ n, 1 ^ s ^ n.
According to the choice of (rn 1,..., fns), the coordinates
x(k),...,x^) (k =1,...,N)
(17)
of every node of the cubature formula (2) differ from the points {2-mr (2jr — = = supp {Km ,2jr n supp {Km ,2jr}, jr = 1,..., 2mr-1, r = 1,...,s, and each of the closed s-dimensional binary parallelepipeds
lm 1 + 1,j1 x ... x lms + 1,js (18)
contains at most one node of the cubature formula (2) (by this fact every binary segment lmr+1,jr = supp {Kmr,jr} contains a projection at most one of node of the cubature formula), jr = 1,..., 2mr, r = 1,..., s. Then the equality (16) can be rewritten as
C2m1 2ms
y(i1,...,is) (q) = 2-m 1-...-ms) y ... y
|j1 = 1 js = 1
N
E Ck Km 1,j1 (x(k M ... k=1
Kms,js \xi
(k)
N 2m1 2ms
2-m 1-...-mE E ... E
lk=1 j1 = 1 js = 1
CkKm 1,j1 (x(^J ... Kms,js [xi )
(19)
1 < i1 < ... <is < n, 1 < s < n. Here we use the fact that the sum
N
Ck nm 1,j1
K x(k) . . . K x(k) Km, j, I xi1 j ... Kms ,js 1 xis
k=1
contains at most one nonzero term for any ordered set (j1,... ,js).
Consider the coordinates (17) of nodes of the cubature formula (2) satisfying the equality
x(k) = 2-mr jr, 1 < jr < 2mr, 1 < r < s.
The following (s + 1) cases are possible for the quantity of such coordinates of the nodes.
u q
x
K
¿1
u q
u q
1. Equality (20) does not hold for any of the coordinates (17) of the nodes (for definiteness, the numbers of such nodes are denoted by k = 1,... ,N1).
2. Only one coordinate in (17) satisfies equality (20) (let k = N1 + 1,..., N2 be the numbers of nodes whose coordinates satisfy this condition).
3. Exactly two coordinates in (17) satisfy equality (20) (to be specific, we assume that the coordinates of the nodes with numbers k = N2 + 1,... ,N3 obey this condition).
s + 1. Equality (20) holds for all s coordinates (17) (let k = Ns + 1,... ,N be the numbers of nodes whose coordinates satisfy this condition).
Moreover, each of the nodes with the numbers k = Nr + 1,..., Nr+1 belongs to exact 2r closed s-dimensional binary parallelepipeds of the form (18), where r = 0,1,..., s, N0 =0, Ns+1 = N.
Given the above, as well as the equality (11), the relation (19) can be rewritten as
( \ _ rt—fh 1— ...-
Efh 1,...,fhs (q) = 2
N1
k = 1
N2
E
k=Ni + l
E (2f 1+"'+fhsck)q +2 E (2fi+'"+fhs — iCk)q +
N3 q N r
+ 4 E (2fh 1+...+™ s—2Ck)q + ... + 2s E (2fh 1+'''+■hís—sCkУJ
k=N2 + 1 k=Ns + l
N1
N2
N3
E Ckq + 21—qYl Ckq + 22(1—q) E Ckq +... + 2s(1—q) E Ckq
k = 1
k=N1 + 1
k=N2 + l
N
E
k=Ns + 1
(21)
1 ^ il < .. . < is ^ n, 1 ^ s ^ n.
Since this reasoning holds not only for (m 1,..., ms), but also for any ordered set (m1,..., ms) such that m1 + ... + ms > m 1 + ... + m s it is true that the value E^'.'.'.^ms (q) does not depend on m1,... ,ms for all (m1,... ,ms) satisfying the inequality m1 + ... + ms > > fh 1 +... + m s. Therefore, sup in the equality (15) reduces to max ,
m1 + '''+ms'^M
whence we obtain the assertion of the lemma.
M^m1 + ...+ms ^ fh 1 + ...+^s
□
Let q be a number related to p by
Let us prove the following theorem.
1 +1 = 1
pq
(22)
Theorem 1. If the cubature formula (2) is exact for any constant, then its error functional satisfy the following relations:
\Sn [f]\< If Ik sup Ef^f (q), f e sp
m1 ,.",ms GN
IISnIk = sup E^f (q).
p m1,. . .,ms GN
If the cubature formula (2) possesses the Haar d-property, then
\Sn [f]\ < If Ik sup Ef^-f (q), f e Sp
m1+."+ms >d
II<NIs. = sup Effi;.... f (q).
p ml+'''+ms >d
(23)
(24)
(25)
q
q
Proof. Let the cubature formula (2) be exact for any constant. By virtue of (4), (5), the inequality (23) follows from (8). Using (23), we obtain:
\\SN||< sup (q).
m\ , . . . ,ms GN
In order to establish that this inequality can not be improved, we use the technique applied in [8]. For M = s, we fix the ordered set (M1,..., Ms), the existence of which was proved in Lemma 2. We introduce the following notation:
Q(il, — ,is) = 2--2
M1-1 _ Ms-1 ■■ 2
N
yjck xmi„i (4fc)) ■■■xms,3s ■
k=1
Then, according to Lemma 2, we have
sup zm^m (q) = ^""m. (q)
m i + . . . +ms^M
'2M1 -1 2Ms-1
£ ■■■ £
j1 = 1 js = 1
N
y e(il '■ ■ ■
^ 31, ■ ■ ■,
k = 1
(i1,...,is)
(27)
Consider the function
2M1-1 2Ms-1
fM^M (xi,...,xn ) = £ ■■■ £ sign ej^j
e(i1,...,is) e31,...,3s
q-1
XM1 ,j1 (xi1 ) ■ ■ ■ Xms ,js (xis ) ,
j 1 =1 j; = 1
1 ^ i1 < ... < is ^ n, 1 ^ s ^ n. For this function, the Fourier-Haar coefficients are given by
co
= 0, (ii,...,is)
i.)J sign jjs
e(i1 ,...,i;) e31, ■ ■ ■ ,3;
q-1
0
mi = Mi,.. ., ms = Ms otherwise.
Then, taking into account the relation (4) and the equality (q — l)p = q, which follows from (22), we have:
A(i1,...,is) ( f (i1,...,is)\ =2 Ap \JM1,..,Ms) =2
At the same time, according to (9),
M1-1 + + M;-1 2 +...+ 2
2M1-1 2Ms -1
Ji1,...,is)
£ ■■■ £
31 = 1 3s = i
(28)
Ö
N
(i1,...,is)
N
x£ CkXMj (xD ■■■Xms„
M1,...,M; (k)
2M1-1 2MS-1
- £ ■■■ £
j1 = 1 js = 1
sign e^j
e(i1'...'is)
31,...,3s
q-1
k=1
(k)
nM1-1 + + Ms-1 - v^ " v^
= -2 2 2 £ ■■■ £
2M1-1 2Ms-1
31 = 1 j; = 1
e(i1'...'i;)
31,...,js
The last relation, combined with (27) and (28), shows that
N
f (i 1 ,...,i s ) fM1,...,Ms
= 2 ^+-+M-
2M1-1 2MS-1
£ ■■■ £
j1 = 1 js = 1
e(i1'...'i;)
31,...,js
2M1-1 2Ms-1
£ ■■■ £
31=1 3s=1
(
i(i1,...,i;) f f (i1,...,i;) \ S (i1 ,...,i; )
AP \JM1,..,Msj SM1,..,Ms
(q).
q
q
p
q
X
q
p
q
X
q
q
X
Note that A
(k1,...,ks) ( f(il,...,is)
M1,...,MS
0 for all ordered sets (k\,... ,ks) = (ii,... ,is). Then
f (^v-jO II _ A(i1,...,is) I f
>Mi,..,Ms llS _ ap U,
Mi,..,Ms!, ana
N
(i1,...,is)
M1,...,Ms
y(i1,...,is) ^M1,..,Ms
(q)
•(i1,...,is)
M1,...,Ms
which implies the equality (24).
If the cubature formula (2) possesses the Haar d-property, then by virtue of its accuracy for Haar polynomials of degree at most d, the equality (9) becomes
— 1 oms — 1
2m1
*n [f]_ -y E E E-E
s = 1 1^i1<...<is^nm1 + ...+ms>d j1=1 js = 1 N
X
k=1
■,(j1,...,js)
(il, .. .,is)x
J2CkXmij (x^) ...Xmsjs (^f) \. k=1 '
Hence, the inequality (8) can be written as
l*N If ]| <E E E 2 ^+-+^ x
s = 1 1^i1<...<is^n m1 + ...+ms>d
|-r>m1 — 1 2ms — 1
E ••• E | jj (i1,...,is)
- j1=1 js = 1
s^',...(q) •
Then the inequality (23) becomes (25). Proceeding as in the proof of the equality (24), we
construct the function ÎmY'^Ms X ,..xn) such that
N
■(i1,... ,is)
M1 ,...,Ms
•(i1,...,is) M1 ,...,Ms
sup s^-m (q),
Sp m1 + ...+ms>d
(29)
where the ordered set (M\,..., Ms) satisfies the following conditions:
Mi + ... + Ms >d,
sM. : M (q) _ sup sm-m (q).
m1 + ...+ms >d
This ordered set exists by virtue of Lemma 2, which is used for M = d + 1. The equality (26) follows from (25) and (29).
Lemma 3. For positive integer m1,..., ms satisfying the inequality
m1 + ... + ms ^ d,
it is true that
Qn Km1;''.;mS (xi1 ,...,xis ) = 1 Kjl.fl (xi1, ... , xis )
(30)
(31)
where 1 ^ i,1 < ... < is ^ n, s _ 1,... ,n, jr _ 1,..., 2m
r- 1
r _ 1, . . . , s.
S
p
m1 ,...,m
X
1
Proof. Since each of the functions Kmi,j1 (xil),..., Kms,js (xis) is a Haar polynomial of one variable and the degrees of these polynomials are mt , . . . , ms respectively, then it follows from (14) that for mt ,...,ms satisfying the condition (30), the function Kj]'.'.',:mml (xil ,...,xis) is a Haar polynomial of degree mt + ... + ms < d of variables xil,... ,xis. Then, by virtue of the accuracy of the cubature formula (2) for the Haar polynomials of degree at most d, the first equality in (31) holds true.
The second equality in (31) follows from the relations (14) and (11), which define the functions
Kmi,...,ms (xi1 , ... , xis ) and Km1 ,j1 (xi1 ), ... , Kms ,js (xis ) . □
Lemma 4. For positive integer l, the following inequality holds:
Om1 —l 2ms—l
S (q) < 2—mi
—mi—... —m
{2m! £
jl = 1 js = 1
... £ Qn
k (ji'-'js) [t
Km1-l,...,ms-r x
-l(xii ,■■■■.
(32)
where 1 ^ ii < ... < is ^ n, mi,..., ms = 1, 2,..., s = 1,... ,n. Proof. Inequality (32) is proved by induction on l.
Applying the triangle inequality, and also taking into account the equality (12) and the positivity of the coefficients at the nodes of the cubature formula (2), we obtain:
N
£ Ck Xmi,ji (x^) ■ ■ ■ Xms,js fâj)
N
J2Ck
k=1
k=1
Kmi,2ji-1 IX'iJ " Kmi,2ji
m1 +1
< 2--2—
2 x
ms,2js-1 [xiJ j Kms,2js [x
„ (k)
(33)
The nonnegativity of the functions Km,j (x) implies the inequality
(x(k^ - Kmr ,2jr (x(k^ < Kmr,2jr — 1 (x^) + Kmr,2jr (^f)
r = 1,... ,s, k = 1,..., N. Then, by virtue of the equalities (13) and (14), it is true that
(k)
„(k)
(k)
(k)
<
<
,2ji-1 (^f) + Kmi,2ji (a^) ■ ■ ■ Kms,2js — 1 (x^) + Kms,2js (x^)
2
mi-1,ji [x(-L ) ■ ■ ■ Kms-1,js
(k)
2s K(ji;-;js) ¡x(k) x(k)
2 Km1 -1,...,ms-1 \xi1 ,...,xis
Combining this with (33) yields
N
"y^, Ck Xmi,ji {^Ik )) ■ ■ ■ Xms,js (~js k=1
(k)
m1 +1
< 2--2—
L + s
Q
N
TV-til^-js")
Kmi-1,
ms — 1
(xii, ■ ■ ■ , xis ) ,
which implies (32) for l = 1.
Based on the induction hypothesis that
s)
(q) < 2—mi
—mi—... —ms+ls — s N
< 2mi — i + 1 2ms — i + i
£ ■■■ £
» j1 = 1 js = 1
Q
N
K (j1,-,js) (x
Km1—l + 1,...,ms—l+1\ xii
xi , . . . , xi
9 1 q
(34)
9 q
x
ms + 1
K
m
K
2
X
we prove (32). The sum on the right-hand side of the inequality (34) can be written as
2m1 — J + 1 2ms — ¿ + 1 q
fcih^-ijs)
m1 — l + 1,...,m.
y - £ \Qn
j1 = 1 js = 1 1
2m1 — l 2ms — l 2j1 2j
_y E - E \Qn
j1 = 1 js = 1 J1 = 2j1-1 Js = 2js-1 Using inequality
-l+1 (Xi1 , . . . , Xis )
(35)
KrrJ1-1+1''1 \X'i1 , ... ,xi
—1+1 (Xi1, . . . ,xis)
m r m ^ q
]aq ^ S E ai } {a
i=1 i=1
and equality (13), we have:
Eaq < E ai\ (ai > 0, i _1,...,M,q> 1)
2j1 2js
E - E \Qn
J1=2j1-1 Js=2js-1
r(J1,...,Js)
im1—l + 1,...,ms—l+1 ^
Km 1 l 11 m l i1 (Xi1 , . . . , Xis )
<
r(JU...,Js)
s-l+1(Xi1 , . . . ,Xis)
2j1 2js
<< Qn E ... y Km1-l+1,..,m
I lJ1 = 2j1-1 Js = 2js-1
2j1 2js
QN E ... Ys Km1-l + 1,J^Xi^ . . .Kms-l+1,Js (Xis)
J1 = 2j1-1 Js = 2js-1
_ <Q
N
(yKm1-l+1,2j1-1{ Xh) +K„n-l+1,2j1 ( Xi^j . . .ÇiKms-l + 1,2js-1( XiJ + Kms—l+1,2js (Xis) J
QN
2 Km1 —l,j1 (Xi1 ) ... Kms —l,js(Xis )
2'SQN
Km1—l,j. .]ms—l(Xi1, . . . ,Xis)
In view of the equality (35) and the last relations, it follows from (34) that the inequality (32) holds true. □
Lemma 5. If the cubature formula (2) possesses the Haar d-property, then
sup Ef1:;;f (q) < 2^1 (2d) — p . (36)
ml+'''+mS >d
Proof. Let (m1 ,...,ms) be an arbitrary fixed set of indices for which the inequality m1 + ... + ms > d holds true. We denote by l the minimal number among all integers L satisfying the condition
m1 + ... + ms — Ls ^ d. (37)
Then the following equality holds:
m1 + ... + ms — ls _ d — r, where r G {0,1,..., s — 1} .
(38)
Applying Lemmas 4 and 3 (by virtue of (37), the condition of Lemma 3 for the lower indices of the Haar polynomial Km—i^m^AXi1,.. .,Xis) is satisfied) and taking into account (22) yields
m.1 —l oms —l \ 1
'2m1—l 2
s^M (q) < 2—m1—...—ms+lsïJ2 .-Y 1
I j1 = 1 js = 1 ) _2—m1—...—ms+lsf 2m1+...+ms—ls \ q _ f 2m1+...+ms—ls
q
q
q
q
q
p
The relations (39) and (38) imply
_ 1 r _ 1 s_ 1 _ 1 1 _ 1
^m^wfrnl (q) < (2d-r) p = 2p (2d) p < 2-p (2d) p < 2— {2d) p ,
whence we obtain the inequality (36). □
Lemma 6. If the cubature formula (2) is exact for any constant, then
sup £m\:\ : : (q) > (2n+t — n — 1)-p n-p. (40)
m1,... ,ms GN
Proof. Consider the function
N1 N2 N3 N
V>(Ct,...,CN) = ^2 Ckq + 2t-q ]T Ckq+22(t-q) Y, Ckq + ... + 2s(t-q) Y. Ckq, (41)
k = t k=N1 + t k=N2 + t k=Ns + t
where the constants Nt, . . . , Ns are defined in the proof of Lemma 2. By virtue of (21), the equality
.. m (q) = te (Ct,C2,...,CN)]q (42)
holds true.
If the cubature formula (2) satisfies the condition
Ct + C2 + ... + Cn = 1(Ci > 0, i =1,2,... ,N),
which follows from the accuracy of (2) for any constant, it is easy to show that the function (41) attains its infimum, which is equal to
[Nt + 2 (N2 — Nt) + 22 (N3 — N2) + ... + 2s (N — Ns)]1-q = = [N + (2t — 1) (N2 — Nt) + (22 — 1) (N3 — N2) + ... + (2s — 1)(N — Ns)]t-q ,
when
Ct = C2 = ... = CN1 = [Nt +2 (N2 — Nt) + 22 (N3 — N2) + ... + 2s (N — Ns)]-t, CN1+t = CN1+2 = ... = CN2 =2 [Nt +2(N2 — Nt) + 22 (N3 — N2) + ... + 2s (N — Ns)]-t, Cn2+i = CN2+2 = ... = CN3 = 22 [Nt +2 (N2 — Nt) + 22 (N3 — N2) + ... + 2s (N — Ns)]-t,
CNs+t = CNs+2 = ... = Cn = 2s [Nt + 2(N — Nt) + 22 (N3 — N2) + ... + 2s (N — Ns)]-t. Then, taking into account (22), we derive from (42)
ZmW'... m (q) > [N + (2t — 1) (N2 — Nt) + (22 — 1) (N3 — N2) + ... + (2s — 1)(N — Ns)]- p >
> [N + (2t — 1) N + (22 — 1) N + ... + (2s — 1) N]- p = (2s+t — s — 1)-p N- p >
> (2n+t — n — 1)-p N-p,
where (mht,..., fhs) is the ordered set chosen in the proof of Lemma 2 (in this case M = s, where M is the parameter from the conditions of Lemma 2). This yields the inequality (40). □
Theorem 2. For the cubature formula (2) exact for any constants, the norm of the error functional satisfies the inequality
IIJnIIs. > (2n+1 — n — 1) — 1 N—1. (43)
If the cubature formula (2) possesses the Haar d-property, then
\Sn [f]\ < 2^ (2d) — 1 If IIsp, (44)
IISnIIs. < 2^1 (2d) — p . (45)
Inequality (43) follows from Theorem 1 and Lemma 6, while inequalities (44), (45) follow from Theorem 1 and Lemma 5.
N
' g(x)f (x) dx ^Y, Ck f (x(k)) , (46)
Remark 1. In [9] one considered the following weighted quadrature formulas possessing the Haar d-property:
1N
Ck f I x^
k=1
where x(k) e [0,1] are the nodes of a formula; Ck are the coefficients of the formula at the nodes (real numbers); and k = 1,..., N. If the weight function g(x) = 1, then the number N of nodes of the quadrature formula (46) satisfies the inequality N > 2d—1. The last inequality follows from a lower estimate for the number of nodes of the quadrature formula (46) possessing the Haar d-property, where g(x) is an arbitrary weight function (see [9]).
Moreover, in [9] all minimal weighted quadrature formulas possessing the d-property were described. In the case of the weight function g(x) = 1, it was proved that the minimal formula is unique: the number of its nodes is N = 2d—1, the nodes of this formula are x(k) = 2—d(2k — 1), and the node coefficients are Ck = 2—d+1 for k = 1,2,..., 2d—1. The norm of the error functional of this formula satisfies the equality (see [10])
II*n Is. = 2—1N— p, (47)
which also follows from the inequalities (43) and (45) for n = 1; a number d related to N by N = 2d—1.
Remark 2. In [12], one constructed the minimal cubature formulas possessing the Haar d-property for d ^ 5:
n1 N / \ f (xux2) dx1 dx2 « V Ck f (x[k),xik)) , (48)
k=1
where (x1k\x<2')^ e [0,1]2 are the nodes of a formula; Ck are the coefficients of the formula at the nodes (real numbers); and k = 1, . . . , N. The number N of nodes of such formulas satisfies the equality
N ={2d — 3 J 2 — +2, d is odd (49)
| 2d — 2 2 +1 +2, d is even, v 7
where d =5, 6, 7,... Then, the norm of the error functional of the minimal cubature formulas (48) possessing the Haar d-property satisfies the inequality
¥NIs. < EN, (50)
where EN J 2 p ("N + ¥VN"f + , d is odd, (51)
[ 2p (N + 2VN - 1) p , d is even.
The inequality (50) follows from the estimate
\\Sn\\sp < 2p(2d)-p,
which was obtained in [16] for the norm of the error functional of arbitrary cubature formulas (48) having the Haar d-property. The number N of nodes of these cubature formulas is defined by (49).
The relations (50), (51) also follows from (45) for n = 2; a number d related to N by (49).
3. Conclusions
In [8], the cubature formulas
r1 f1 1 N / \
Jo ■■■ Jo f(xi ,...,xn) dxi... dxn « (52)
k=i
with nodes [xt^, • • •, x<n ^ G [0,1}n (k =1,..., N) were considered that form PT-nets, i.e., nets that consist of N = 2V nodes and satisfy the following condition: each binary parallelepiped of volume 2T-v contains 2T net points (v > t). For such formulas with a function f from Sp, the following upper estimate for the norm of the error functional was proved in [8]:
\\Sn|U < 2N-p. (53)
It is easy to see that for n = 1 and n = 2 PT-nets with an arbitrarily large number N = 2V of nodes exist for any t = 0,1,2,... Therefore, in the one- and two-dimensional cases, the constant multiplier on the right-hand side of (53) takes the least value at t = 0, and estimate (53) for the cubature formulas (52) with nodes forming P0-nets in the one-dimensional case is written as
\\SN\\s. < N-p, (54)
while in the two-dimensional case this estimate is written as
\\5N\\sp < 2pN-p. (55)
It was proved in [8] that cubature formulas (52) with 2d nodes forming P0-nets have the Haar d-property. Therefore, the estimate (45), which is obtained in the present paper, is a generalization of the estimate (53) to the case of arbitrary cubature formulas possessing the Haar d-property.
Moreover, for any cubature formula (52) with a function f G Sp, it was established in [8] that the norm of the error functional satisfies the lower estimate
\\5N\\Sp > N-p.
Hence, the cubature formulas (52) with the nodes forming PT-nets have the best convergence rate of 5n in the norm, which is equal to N-p as N ^ to.
The relations (43), (47), (50), (51) imply that for minimal formulas possessing the Haar d-property in the one- and two-dimensional cases HSNII s. x N— p as N ^ x>.
Comparing the values on the right-hand sides of the relations (47) and (54), as well as (50) and (55), we conclude that the upper bounds for the HSNIs. in the case of minimal quadrature formulas (46) with the weight function g(x) = 1 and the minimal cubature formulas (48) with the d-property are less than the upper bounds for this value in the inequalities (54) and (55), respectively, i.e., the upper bounds for the norm of the error functional of formulas with nodes forming the P0-net in the one- and two-dimensional cases.
In addition, the quadrature formula (46) with the weight function g(x) = 1 and the number N = 2d—1 of nodes, as well as the cubature formula (48) with the number N of nodes satisfying the equality (49), being the minimal formulas of approximate integration, provide the best pointwise convergence of SN [f ] to zero as N ^ x>.
References
[1] V.I.Krylov, Approximate Calculation of Integrals, Nauka, Moscow, 1967 (in Russian).
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Об оценках погрешности на пространствах Sp кубатурных формул, точных для полиномов Хаара
Кирилл А. Кириллов
Сибирский федеральный университет Красноярск, Российская Федерация
Аннотация. Получены верхняя и нижняя оценки нормы функционала погрешности обладающих
d-свойством Хаара кубатурных формул на пространствах Sp в те-мерном случае.
Ключевые слова: d-свойство Хаара, погрешность кубатурной формулы, пространства Sp.
