On error estimates for weighted quadrature formulas exact for Haar polynomials Текст научной статьи по специальности «Математика»

УДК 517.518.87

On Error Estimates for Weighted Quadrature Formulas Exact for Haar Polynomials

Kirill A. Kirillov*

Institute of Space and Information Technologies, Siberian Federal University, Kirenskogo, 26, Krasnoyarsk, 660074

Russia

Received 04.01.2013, received in revised form 14.03.2013, accepted 20.04.2013 On the spaces Sp, estimates are found for the norm of the error functional of weighted quadrature formulas. For quadrature formulas exact for constants a lower estimate of || is proved, and for quadrature formulas possessing the Haar d-property upper estimates of the ||S, are obtained.

p

Keywords: Haar d-property, error functional of a quadrature formula, function spaces Sp.

Introduction

The problem of constructing and analyzing cubature formulas that integrate a given collection of functions exactly has been earlier considered mainly in the cases when these functions are algebraic or trigonometric polynomials. The approximate integration formulas exact for finite Haar sums can be found in the monograph [1], the accuracy of approximate integration formulas for finite Haar sums was used there for deriving the error of these formulas.

A description of all minimal weighted quadrature formulas possessing the Haar d-property, i.e., formulas exact for Haar polynomials of degree at most d, was given in [2]. In [3] the estimates of the norm of the error functional for quadrature formulas exact for Haar polynomials were proved on the spaces Sp and Ha in the case of the weight function g(x) = 1. In [4] the estimates of the norm of the error functional for minimal quadrature formulas exact for Haar polynomials were obtained on the spaces Sp in the case of the weight function g e Lx[0,1].

In the two-dimensional case, the problem of constructing cubature formulas possessing the Haar d-property was considered in [5-9]. Estimates of the norm of the error functional for cubature formulas on the spaces Sp and Ha were obtained in [10,11].

The results obtained in [3] are extended in this paper to the case of weighted quadrature formulas on the spaces Sp. In the case of the weight function g e Li[0,1], a lower estimate of the norm of the error functional SN is derived for quadrature formulas exact for any constant. In the cases of the weight function g e LTO[0,1] and g e Lq[0,1] (p-1 + q-1 = 1), upper estimates of the norm of the error functional SN are proved for quadrature formulas possessing the Haar d-property. It is shown that in the case of N x 2d with d ^ <, the value \\SN \\ for the formulas

i

under study has the best convergence rate to zero, which is equal to N-p with N ^ <.

* kkirillow@yandex.ru © Siberian Federal University. All rights reserved

1. Basic definitions

In this paper, we use the original definition of the functions Xm,j (x) introduced by A. Haar [12], which differs from the definition of these functions used in [1].

Dyadic intervals Zm,j are the intervals with their endpoints at (j — 1)/2m-1, j/2m-1, m = 1,2,..., j = 1,2,..., 2m-1. If the left endpoint of a dyadic interval coincides with 0, then we consider this interval to be closed on the left. If the right endpoint of a dyadic interval coincides with 1, then we consider this interval as closed on the right. The remaining intervals are considered open. The left (right) half of Zm,j (without its midpoint) we denote by Zm ■ (Z++ j).

It is convenient to construct the Haar system of functions in groups: the mth group contains 2m-1 functions Xm,j(x), where m =1, 2,..., j = 1, 2,..., 2m-1. The Haar functions Xm,j(x) are defined as:

2 ^,

-2 ^,

Xm,j (x) = <

if x e Zm ,j,

. if x e Z++ ,j,

0, if x e [0,1] \ Zmj,

\[Xm,j(x — 0) + xm,j(x + 0)], if x is an interior point of discontinuity of the

function Xm

with lm,j =

j - 1 j

j

m -1

, m = 1, 2,..., j = 1, 2,..., 2m 1. The Haar system of functions

2m-1 ' 2m-1

includes the function x1(x) = 1 too, which is outside of any group. The Haar polynomials of degree d are by definition the functions

d 2m-1

Pd(x) = aoX1(x) + a<m^Xm,j(x),

m=1 j=1

where d = 1, 2,..., a0, a^ G R, m = 1, 2,..., d, j = 1, 2,..., 2m

od-1 2 2

-1

and

E(odj0 =0.

j=1

By the 0-degree Haar polynomials we understand real constants. We consider the following quadrature formula

, 1 N

I[f] = g(x)f (x)dx ^Cif (x(i)) = Qn[f], Jo i=i

(1)

where x(i) e [0,1] are the nodes, the coefficients Q at the nodes are real and satisfy the inequalities

Ci > 0, (2)

i = 1, 2,..., N, the functions g(x), f (x) are defined and summable on [0,1].

We denote the error functional of the quadrature formula (1) by SN [f ] so that

Sn [f ]

N

E

i=i

Cif (x(i)) - g(x)f (x) dx.

0

(3)

The quadrature formula (1) is said to possess the Haar d-property (or just the d-property) if it is exact for any Haar polynomial Pd(x) of degree at most d, i.e., QN[Pd] = I[Pd].

1

2. Estimates for the norm of the error functional for weighted quadrature formulas

We recall the definition of the classes Sp introduced by I.M. Sobol' [1]. Let p be a fixed number with 1 ^ p < For an arbitrary positive A one can define the class Sp(A) as the set of functions f (x) on [0,1] that can be represented by a Fourier-Haar series

o 2m— 1

f (x) = c01} (x) (4)

m=1 j = 1

with real coefficients c^, c^ (m =1, 2,..., j = 1, 2,..., 2m—1) satisfying the condition

|cm

2

AP(f )=E2^[E < A. (5)

m=1 j=1

It is proved in [1] that J2 Sp(A) with the norm

A>0

llf Ik = Ap(f), (6)

forms a linear normed space, which is denoted by Sp. All the functions f(x) that differ by constant terms are regarded as a single function.

To establish an estimate for the norm of the error functional of the quadrature formula (1) we need to recall the definition of the space LO[0,1] [13]. It consists of all measurable almost everywhere finite functions g(x), and for each function there exists a number Cg such that |g(x)| < Cg almost everywhere. We call such functions essentially bounded on [0,1]. For a function g e LO[0,1] one defines the proper (essential) supremum of its absolute value

ess sup |g(x)|

x£[0,1]

as the infimum of the set of numbers a e R such that the measure of the set

{x e [0,1] : |g(x)| > a}

is zero. LO[0,1] is a linear subset in the set of measurable almost everywhere finite functions. The norm on LO[0,1] can be introduced as

l|g|lwo,1] =esssup |g(x)|. (7)

x£[0,1]

Lemma 1( [2]). Let m be a fixed positive integer. The functions

( 2m, if x e lm+1,j,

Km,j (x) = < 2m—1, if x e lm+1,j \ 1m+1,j, (8)

(0, if x e [0,1] \ im+j,

j = 1, 2,..., 2m, are Haar polynomials of degree m and they form a basis in the linear space of Haar polynomials of degree at most m.

Lemma 2. For all m = 2, 3,..., j = 1, 2,..., 2m—1,

Xm,j (x) = 2-^ [fcm,2j—1 (x) - Km,2j (x)] ,m = 1, 2, ... j = 1, 2,..., 2m—1, (9)

Km,2j-1(x) + Km,2j (x) = 2«m- 1 j (x). (10)

OO

The definition of the Haar functions and relation (8) imply (9), (10). Fix p > 1, then let q be such that —|— = 1.

p q

Lemma 3. For all nonnegative integer k ^ m (m £ N),

2 m 2 m— k

E(/ |g(x)| dx + 2-mQ[«mj]) 9 < E(/ |g(x)| dx + 2-m+kQ[Km-k,j ]) '. (11)

j=1 j = Jlm — k + 1,j

j=1 Jim — k + 1,j

Proof. Inequality (11) is proved by induction on k. For k = 0 it becomes an equality. Based on the induction hypothesis that

2m 2m—k + 1

E(/ |g(x)| dx + 2-mQ[Kmj])9 < E (/" |g(x)| dx + 2-m+fc-1Q[Km-fc+iij])9,

j=1 J 1 m+1 ,j j=1 ^ ^m— k + 2,j

j =1 'm—fc + 2,j

we prove (11). For a, b > 0 and q > 1, it is easy to see that

a9 + b9 < (a + b)9. In view of (13) and (10), it follows from (12) that

(12) (13)

2 m— k + 1

E (/ lg(x)l dx + ^^

j=1 V^'m — k + 2,j

< E

j=1

|g(x)| dx +

+ I / |g(x)| dx +

' lm — k + 2,2j —1

Q[«m-k + 1,2j-1]

2m-k+1

2m

<

E

|g(x)| dx +

Q[Km-k+1,2j] 2m-k+1

Q[Km— k,j ] ^

|

j=1 ^ 'm— k + 1,j

2

t,-k

Inequalities (12) and (14) imply (11). Let

(m)

where m = 1, 2, 3,...

Eh

j=1

g(x) dx + g(x) dx + 2 2 Q[xm,j]

(14)

(15)

Lemma 4. If f e Sp, g e L1 [0,1], then for the quadrature formula (1) all (m) are bounded, and

~ 2d

(m)| < 2 £(/ |g(x)| dx

j = 1 Jld + 1,j

where m = d + 1, d + 2,...

Proof. In view of (9) and (10), it follows from (15) that

(16)

(m)]9 < E i — |g(x)| dx + / |g(x)| dx + 2-m|Q[Km,2j-1] - Q[«m,2j]|

j=1

h+

<

< E [ / |g(x)| dx + 2-mQ[Km,2j-1 + Km,2j]|1 * == E i / |g(x)| dx + 2-m+1 Q[Km-1,j]

j=1

j=1

9

9

9

9

m,j

m,j

1

2

9

m, j

—1

1

9

By Lemma 1 and relations (2), we have

' ld + 1,j Therefore

1 N

/ g(x) dx = 2-d/ g(x)Kdj(x) dx = (x(i)) > 0, j = 1, 2,..., 2d.

Jid+i,j Jo i=i

g(x) dx ^ 0.

(18)

ld + 1,j

Using Lemma 3, Lemma 1 and inequality (18), for m = d + 1, d + 2,... we obtain

E

j=i

2d j=i

Jm |g(x)| dx + 2-m+1Q[Km-U]

/ |g(x)| dx + / g(x) dx l d+1 l d + 1

q 2d

j=i q 2d

j=i

|g(x)| dx + 2-dQ[KdjJ-]

<

2 j |g(x)| dx

l d + 1

(19)

Inequalities (17) and (19) imply (16). It follows from inequalities

(m)| < max J (1), (2),..., (d), 2

E(/ |5(x)| dx

j=i J ld+ 1,j

m = 1, 2,. ..

that all (m) are bounded. □

Lemma 5. If g G Li[0,1], then for the norm of the error functional of the quadrature formula (1) exact for any constant we have

ll^w Us* = sup (m).

If, in addition, the quadrature formula (1) possesses the Haar d-property, then

ll^w Hsp = sup ^q (m).

d<m<o

(20)

(21)

Proof. The series (4) is substituted into (3). Since the quadrature formula (1) is exact for any constant, it follows that

œ 2m

¿n f ] = EE

,(j)

m=i j=i

■ N i

E CiXm,j (x(i)) -/ g(x)Xm,j (x) dx

o

(22)

By virtue of the definition of Haar functions, it follows from (22) that

°° 2m I r c c

¿n f] = E 2 ^ E Ie™ - - g(x) dx + g(x) dx + 2-^ Qixm.j ]

_1 -_1 I _ Jl— J l +

m=i j=i

(23)

Since the series (23) converges uniformly, it is true that

|«nf]| <E 2^ E

=i j=i

,(j)

g(x) dx + / g(x) dx + 2 2 Q[xm,j]

(24)

q

2

d+1, j

q

d

2

q

l

m

•m.j

m, j

2

oo

m

Applying the Holder inequality to the expression in (24), we obtain

l*N [f ]| <E 2

2

E ij

j=i

(m) }.

Then, in view of (5) and (6), it follows from (25) that

l<*N [f]l < llflk sup (m).

(25)

(26)

In order to establish that inequality (26) can not be improved, we use the technique applied in [1]. Let m = m0 e N be a fixed number. We consider the function

fmo (x)

X sign

2mo —1

E<

j=i

g(x) dx + g(x) dx + 2 02 Q[Xmo,j]

g(x) dx + / g(x) dx + 2 02 Q[xmo,j] Xmo,j(x) . J

mo '3

q-i

The Fourier-Haar coefficients of this function are

c(j)

g(x) dx + g(x) dx + 2 2 Q[Xmo,j]

q-i

X sign

g(x) dx + / g(x) dx + 2 2 Q[Xmo,j]

h+

mo '3

, j = 1, 2,..., 2mo-i,

c0O) =0, c^ =0, m G N \ {mo}, j = 1, 2,..., 2m-i. Therefore, (23) implies

2mo-

mo — 1 *-^

|<*N [/mo ]| = 2^ E j=1

g(x) dx + g(x) dx + 2 2 Q[Xmo,j]

h+ m0'j

= ll/mo lisp ^q (mo).

(27)

Since the function fm0 (x) exists for any m0 e N, relation (27) implies (20). It follows from Lemma 4 that the norm of the error functional is finite.

Let the quadrature formula (1) possess the Haar d-property. Since it is exact for Haar polynomials of degree at most d, in the relations (22), (23) and (25) the lower index in the sum over m is equal to d + 1. Therefore, inequality (26) turns into [f]| < ||f ||Sp sup (m).For

d<m< ^

any m0 e N there exists a function fm0(x) satisfying (27), this implies (21). □

Let 1

G = I g(x) dx. (28)

0

Theorem 1. If g e L1[0,1], then for the norm of the error functional of the quadrature formula (1) we have the lower estimate

||5n ||sp > 2-p GN-p, (29)

where the constant G is defined by (28).

Proof. Let £ be a number satisfying the condition

0 < e < min •! —

1<i<N 2

1

X

mo,j

mo 'j

mo 'j

X

mo 'j

mo '3

mo '3

q

mo 'j

There exists a number m0 e N such that for any m > m0 the following inequalities hold:

g(x) dx

< -, j = 1, 2,..., 2'

2' 7 ' ' '

2-1

Let m1 be the minimal number among all numbers m satisfying the following condition:

each of the segments lm+1,1,..., lm+1j2m contains at most one node of formula (1).

2j — 1

If every node of the quadrature formula (1) differs from the points —^—, j = 1,2,..., 2m1-1, we set m2 = m1. Otherwise, we set

m2 = 1 + max{m : there exists xr = j-jr e {1, 2,..., 2m-1}}.

Let m' = max{m0, m2}. Then for all m ^ m' the following three conditions are satisfied:

for all j = 1, 2,..., 2'

-1

j g(x) dx — j g(x) dx

(31)

- each of the segments lm+1,j contains at most one node of the quadrature formula (1), j = 1,2,..., 2m,

2 j — 1

- every node of the quadrature formula (1) differs from the points 0m—, j = 1,2,..., 2m—1

In view of (9), it follows from (15) that

'2"

2m

(2m -1 N

^q (m')=<E—/ g(x) dx + g(x) dx + 2-m'^Ci (

I j—1 JI / . J , . i — 1

V. j — 1 m' ,j m' ,j i— 1

Km',2j-1 (x(i)) — Km',2j (x(i))

1

q 1 q

(32)

It follows from the definition of the number m' that the nodes of the quadrature formula (1) differ from the points

2 j — 1

2m'

| 2j2- ^ = supp (Km',2j-1}n supp (Km',2j },

and each of the segments lm'+1,j contains at most one node of formula (1). Then relation (32) can be rewritten as

(m') = jE ±2-m' E CiKm',3(x(i)) — (£ g(x) dx — J++ g(x) dx) | .

\ j =1 i=1 m',j m',j y

In this formula we choose the plus sign if x(i) e lm' j, and the minus sign if x(i) e l++' j. Taking into account (30) and (31), it follows from (33) that

(33)

^q(m') = < E

j—1

N

2-m'E CiKm,,j (x(i)) —

±

g(x) dx — / g(x) dx

{2m j—1

N

2-m'E CjKm',j (x(i))

(34)

Let N1 be the number of nodes of the quadrature formula (1) that coincide with the points | j7 | (j = 1,..., 2m — 1). To be specific, we denote them by x(1),... ,x(Nl). Since every

m, j

q

q

"i ,j

"i ,j

q

q

segment Zm'+1j- contains at most one node of the formula (1), it follows from (34) that

( N W , C 1

*q (m') > ]T Ciq + 2^ C . (35)

U=Ni + 1 i=1 ^ ' J

Since the quadrature formula (1) is exact for any constant, we have

C1 + C2 + ... + Cn = G, (36)

where the constant G is defined by (28). Because of (2), it follows from (36) that G > 0.

If the quadrature formula (1) satisfies the conditions (36) and (2), it is easy to show that the function

/c \ q N

¥>(c1,c2,...,cn)=2£ ^ + £ Ci

i=1 ^ ' i=Ni + 1 attains its infimum, which is equal to Gq(N + N1)1-q, when

= ... = CN

It follows from (35) that

(m') > G(N + N1)-1 > 2-pGN-p. (37)

—1 = —2 = ... = —Ni = 2G(N + Ni)-1, —Ni + 1 = —Ni + 2 = ... = —n = G(N + Ni)-1.

The relations (20) and (37) imply (29). □

Let 1

Go = [ |g(x)| dx. (38)

Jo

Theorem 2. If g £ LTO[0,1], then the norm of the error functional of the quadrature formula (1) possessing the Haar d-property satisfies the estimate

/ \ 1 i 1 lisp < 2\2-dj PGoq||gH£_[o,1], (39)

where the constant G0 is defined by (38). Proof. In view of (7) and (38),

2'

l5(x)l dx) = E (/ lff(x)l dx) / lff(x)l dx

j=1 d+1'3 j=1 L d+1'3 d+1 'j

< (2-dl|gHL_[o,i])q-i E / lg(x)l dx < (2-dygyi^[o,i])q-1Go.

j=1 1 d + 1'3

(40)

The relations (21), (16) and (40) imply (39). □

Theorem 3. If g G [0,1], then the norm of the error functional of the quadrature formula (1) possessing the Haar d-property satisfies the estimate

1

||5nlisp < 2(2-d) p ||gyL,[o,i]. (41)

Proof. Using the Holder integral inequality, we obtain

/ |g(x)| dx < [2-dyU |g(x)|q dx)\ j = 1, 2,..., 2 .

'id + lj Jld + l,j

(42)

It follows from (16) and (42) that

(m)| < 2

■ 2d

£(2

j=1

q

p I

ld + 1,,

|g(x) |q dx

22

1

|g(x)|q dx

2(2^) p\\g\\Lq[o,!].

The relations (43) and (21) imply (41).

(43) □

Conclusions

In [14] I.M.Sobol' establishes a quadrature formula with the weight function g(x) > 0 and positive coefficients at the nodes that satisfy the following condition

N 1

y Ci = / g(x) dx = G. i= 1 Jo

The error functional SN of that formula can be estimated as follows [14]:

GN-p < \\SN|L. < 2GN-p,

p

and, therefore, \\ JN\\S» x N-p when N ^ œ. Sp

It follows from Theorems 1, 2 and 3 that for g G Lœ[0,1] and g G Lq [0,1] in the case of N x 2d with d ^ œ the quadrature formula (1) possessing the Haar d-property has the best rate of convergence \\JN\\Sp to zero, which is equal to N-p when N ^ œ.

In particular, the minimal weight quadrature formulas constructed in [7] satisfy the condition N x 2d when d ^ œ. At the same time these formulas, being the minimal formulas of approximate integration, provide the best pointwise convergence of SN[f ] to zero as N ^ œ.

1

o

References

[1] I.M.Sobol', Multidimensional Quadrature Formulas and Haar Functions, Nauka, Moscow, 1969 (in Russian).

[2] K.A.Kirillov, M.V.Noskov, Minimal quadrature formulas accurate for Haar polynomials, Comput. Math. Math. Phys., 42(2002), no. 6, 758-766.

[3] K.A.Kirillov, Estimates for the norm of the error functional of quadrature formulas exact for Haar polynomials, J. of Siberian Federal University, Mathematics & Physics, 4(2011), no. 4, 479-488 (in Russian).

[4] K.A.Kirillov, On the error estimates of weighted minimal quadrature formulas exact for Haar functions, Vychislitelnye tekhnologii, special issue, 11(2006), 44-50 (in Russian).

[5] K.A.Kirillov, Minimal cubature formulas exact for Haar polynomials in R2, Problemy matematicheskogo analiza, Krasnoyarsk, Krasnoyarskii Gosudarstvennyi Tekhnicheskii Uni-versitet, 6(2003), 108-117 (in Russian).

[6] K.A.Kirillov, Lower estimates for node number of cubature formulas exact for Haar polynomials in two-dimensional case, Vychislitelnye tekhnologii, special issue, 9(2004), 62-71 (in Russian).

[7] K.A.Kirillov, Construction of minimal cubature formulas exact for Haar polynomials of high degrees in two-dimensional case, Vychislitelnye tekhnologii, special issue, 10(2005), 29-47 (in Russian).

[8] M.V.Noskov, K.A.Kirillov, Minimal cubature formulas exact for Haar polynomials, J. of Approximation Theory, 162(2010), no. 3, 615-627.

[9] K.A.Kirillov, Minimal cubature formulas exact for Haar polynomials of low degrees in two-dimensional case, Vestnik Krasnoyarskogo Gosudarstvennogo Agrarnogo Universiteta, 2012, no. 10, 7-12 (in Russian).

[10] K.A.Kirillov, M.V.Noskov, Error estimates in Sp for cubature formulas exact for Haar polynomials in the two-dimensional case, Comput. Math. Math. Phys., 49(2009), no. 1, 1-11.

[11] K.A.Kirillov, On error estimates for cubature formulas exact for Haar polynomials, Vestnik Sibirskogo Gosudarstvennogo Aerokosmicheskogo Universiteta, (2012), no. 2, 33-36 (in Russian).

[12] A.Haar, Zur theorie der orthogonalen funktionensysteme, Math. Ann., 69(1910), 331-371.

[13] L.V.Kantorovich, G.P.Akilov, Functional Analysis, Nauka, Moscow, 1984 (in Russian).

[14] I.M.Sobol', Weighted quadrature formulas, Siberian Math. J., 19(1978), no. 5, 844-847.

Об оценках погрешности весовых квадратурных формул, точных для полиномов Хаара

Кирилл А. Кириллов

Для весовых квадратурных формул получены оценки нормы функционала погрешности на пространствах Sp — нижняя оценка величины ||S* для формул, точных на константах, и верхние оценки ||Jn||s* для формул, обладающих d-свойством Хаара.

Ключевые слова: d-свойство Хаара, функционал погрешности квадратурной формулы, пространства функций Sp.