Optimal cubature formulas for Fourier coefficients of periodic functions in Sobolev spaces

Applied Mathematics

Optimal cubature formulas for Fourier

coecients of periodic functions

in Sobolev spaces

Haydar Bulgak

1

, Vladimir Vaskevich

2?

1 Research Centre of Applied Mathematics, Selcuk University, Konya, Turkey e-mail:bulgk@karatay1.cc.selcuk.edu.tr

2 Sobolev Institute of Mathematics, SB RAS, Novosibirsk, Russia e-mail:vask@math.nsc.ru

Received: August 03, 2000

Summary.

The purpose of the paper is to discuss methods for con-structing weighted cubature formulas in Sobolev spaces with periodic members.These formulas are intended to approximatethe calculation of Fourier coecients of functions under consideration. The explicit formulas for the weights and the errors of optimal cubature formu-las are obtained by techniques from functional analysis and partial dierential equations.

Key words:

Fourier coecients, optimal cubature formulas, dif-ference operators of Sobolev type, the error estimations, the Filon quadrature formula

Mathematics Subject Classication (1991): 65D32, 65D30, 41A55

1. Introduction and statement of the results

Let x = (x1:::xn)

2 Rn, H be a real square nn matrix, and

detH >0. To the matrix H we assign itsfundamental parallelepiped

0 by putting

0=

fx2Rn:x=Hy 0yj <1 j = 12:::ng:

? This work was supported by T UBITAK within the framework of a NATO-PC Advanced Fellowships Programme and by RFBR Grant No. 98-01-00760.

We consider the problem of computing the Fourier coecients a'] = Z 0 ei2H ;1x '(x)dx

of an (unknown) continuous real function'(x). HereH is the adjoint matrix to H H ;1 is the inverse matrix to H  = (

12:::n)

is an integer column-vector (i.e., having integer entries),  6= 0 and

H ;1xdenotes the inner product of the vectorH ;1and the vector

x. If H is an orthogonal matrix, thenH ;1 =H. The domain 0 of

integration and the weight function ei2H ;1x

are related by the equality Z 0 ei2H ;1x dx=Z Q e i2ydy= n Y j=1 1 Z 0 ei2 jyjdy j = 0

withQ the unit cube inRn_.

Given a positive integerM, we consider linear rules of the type

Q'] = X

hH2 0

c ]'(hH ) h= 1=M

for the approximation ofa']. Naturally, we assume that the values

'(hH ) with hH 2 

0 are known. The sum Q'] has N terms

N = 1=hn_{ and N is called the number of formula nodes.}

To each rule Q'] we assign the error (1:1) (l') Z 0 ei2H ;1x '(x)dx; X hH2 0 c ]'(hH )

with '(x) a member of some Hilbert space X = X(0). The value

of l(x) is determined at every function '(x) in X only in the case when '(x) is continuous. We therefore naturally require that X be embedded inC(0), the space of functions that are continuous on0.

Moreover, we require that the embedding ofXtoC(0) be compact.

Consequently, the errorl(x) is a bounded linear functional onX i.e.,

l(x)2X , the dual space.

We dene

(QXN) =jjljX jj= sup jj'jXjj1

fja'];Q']jg

and have the error bounds

Clearly,(QXN) is a measure of the quality of the rule Q and the following number is of the special interest:

opt_{(XN) = inf}

Q (QXN)

where the inf is taken over all considered Q'] (that means over all weights fc ] j hH 2 

0

g). Following Brass (1991) we look at

opt_{(XN) as a measure for the costs (or the complexity) required}

for the computation of a Fourier coecient of a function inX. The rule Qopt _{such that (Q}opt_{XN) =}opt_{(XN) is called}

an X-optimal cubature formula for the computation of a Fourier co-ecienta']. We now turn to the following

Problem 1.

Given a number of nodes N, nd the X-optimal cuba-ture formula of the form (1.1).

In this paper, we solve Problem 1 in case of Sobolev spaces with periodic members. The exact denitions are as follows.

Let the function '(x) with domainRn _{have all derivatives up to}

order klocally integrable in Rn _{with the integral}

Z  X jj=k k! !jD'(x)j 2dx= jj'jL (k) 2 () jj 2

nite for every bounded domain . Also assume that 2k > n, and

'(x) is a periodic function with periodic matrixH, i.e., (1:2) '(x+H ) ='(x) 8 2Z

n_:

Here and in the sequel Z denotes the set of integers. Thus, is an integer column-vector.

We denote the Sobolev space composed by the function with pre-scribed properties by Wf

(k)

2 (H) (cf. Sobolev and Vaskevich (1997)).

Given a positive integerm > n=2 and the numbersak,k= 01:::,

such that 0< a0 <+ 1 a 1 =a2 =:::=am;1= + 1 0< ak <+1forkm

we consider the collection of all functions'='(x) such that' is a member ofWf (k) 2 (H) for 8km, and 1 X k=0 jj'jL (k) 2 ( 0) jj 2 a2 k <+1:

We denote this set by Wf (a k ) 2 (H). The norm of ' 2 f W(a k ) 2 (H) is dened as follows jj'j f W(a k ) 2 (H) jj= 1 X k=0 jj'jL (k) 2 ( 0) jj 2 a2 k : The space Wf (a k )

2 (H) is called the periodic Sobolev space of innite

order (cf. Dubinski (1980)). Fixing the sequence (ak), we will write

e X(H), or simplyXe, instead ofWf (a k ) 2 (H).

Each function from the set

(1:3) fei

2H ;1x

j 2Zng

satises (1.2). We assume that all exponents from (1.3) are the mem-bers ofXe(H). Hence, we have

(1:4) A0 ]  1 X k=0 (2 jH ;1 j) 2k a2 k <+1 8 2Zn withjH ;1

jthe Euclidean norm of the vectorH

;1 . Moreover, the equality holds jj'j e X(H)jj 2= 1 X k=0 jj'jL (k) 2 ( 0) jj 2 a2 k = 1jHj X  A0 ] ja' ]j 2

with '(x) a member of Xe(H) and a

' ] a Fourier coecient of '.

The space Wf (a

k )

2 (H) is a Hilbert space with orthogonal basis (1.3).

If ak !+1 fork = m+ 1m+ 2:::, then e

X(H) change into the Sobolev space Wf (m) 2 (H) of nite order. If a 0 ! +1, am = 1, and ak ! +1 for k = m + 1m+ 2:::, then e

X(H) change into the Sobolev space Le

(m)

2 (H) (see, e.g., Sobolev and Vaskevich (1997)).

Theorem 1.

There exists a unique Xe(H)-optimal cubature formula

of the form (1.1) with the given number of nodes N = h;n. The

weights of this cubature formula and the entire functionA0 ]dened

by (1.4) are related by (1:5) c() 0  ] =h njHj " 1 + X 6=0 A0] A0 ;=h] # ;1 ei2h

with , the members1 of Zn and hH 2 

0. The series in (1.5)

converges absolutely. The norm of the optimal error l()

0 (x) depends on, N,H, and(ak) as follows (1:6) jjl () 0 j e X (H)jj 2= jHj  X 6=0 1 A0 ; =h]  " 1 + X 6=0 A0] A0 ; =h] # ;1 :

Theorem 1 was presented in Bulgak and Vaskevich (2000) without the proof. If ak ! +1 for k = m + 1m+ 2:::, then (1.4){(1.6)

change into more simple equalities and we can use them for the members of the Sobolev spaces with the nite smoothness.

LetXe(H) =Le (m) 2 (H), i.e.,A 0 ] = (2 jH ;1 j) 2m. If= 0, then, by Theorem 1, we have c(0) 0  ] =h njHj jjl (0) 0 j e L(m) 2 (H) jj 2= j 0 j h2m (2 )2m X 6=0 1 jH ;1 j 2m:

These are known results onLe (m)

2 (H)-optimality of \rectangle"

cuba-ture formulas and the presentation of theLe (m)

2 (H)-norm of its error

(see, e.g., Sobolev and Vaskevich (1996, 1997)). Let0= 02 ] and

!(x) = eikx_{. If k is an integer, then the explicit formula for the}

weights of the Le (m)

2 02 ]-optimal quadrature formula was obtained

in Babu ska, Pr!ager and Vit!asek (1966).

The equivalent presentation for the weights of Le (m)

2 (H)-optimal

cubature formulas was presented in Shadimetov (1999).

The Fourier coecients of a function with one real variable are frequently calculated by the Filon quadrature formula (see, e.g., Fi-lon (1928)). The explicit error estimates of the FiFi-lon quadrature for-mula are given by Petras (1990) and K"ohler (1993). From Theorem 1 it follows that in case of the Sobolev spaces with periodic members the error of the Xe-optimal quadrature formula is less than the error

of the quadrature Filon formula.

2. Existence and Uniqueness of the

f

X

(

H

)-optimal

cubature formulas

Let the function !(x) with domain Rn _{be locally integrable in R}n_

and let

!(x+H ) =!(x) 8 2Zn: 1 We will adopt this agreement on summation variables ,

in similar sums in the sequel.

We consider the error (2:1) (l') Z 0 !(x)'(x)dx; X xk 2 0 ck'(xk) where k= 1:::N. If a0

!+1, then we assume that the value of

l(x) on each constant functions is zero, i.e.

X xk 2 0 ck = Z 0 !(x)dx: Hence,l(x) is a member ofXe (H).

Theorem 2.

There exists a unique Xe(H)-optimal cubature formula

of the form (2.1).

Before launching into the proof of Theorem 2, we establish the properties of the auxiliary function

(2:2) E e X(x) = 1jHj X  1 A0 ] ei2H ;1 x:

The value A0 ] is nite in view of (1.4). If a0 = +

1, then we agree

to lead summation in (2.2) over 6= 0. The series in (2.2) and all its

derivatives converge absolutely and uniformly in 0. To make sure

of this, it suces to use the inequalities

A0 ] (2 jH ;1 j) 2l a2 l  8l0: Thus,E e

X(x) is an innitely dierentiable real-valued function which

satises (1.2).

If 0 = 01],am = 1, and ak

!+1for k= 0m+ 1m+ 2:::,

then E e

X(x) change into the function which is proportional to the

Bernoulli polynomialB2m(x) of degree 2m i.e., the \limit" equality E e X(x) = ( ;1)m +1 (2m)! B2m(x) x 201] holds.

Lemma 1.

The functionE e

X(x)is a member ofXe(H). For each error

l(x) of the form (2.1) the equalities

(2:3) (l') = (ul')Xe

8'2 e

X(H):

hold. Hereul(x) is a member of Xe(H) which is dened by

(2:4) ul(x) = Z 0 !(y)E e X(x;y)dy; X xk 2 0 ckE e X(x;xk):

Proof. By the denition of E e X(x), we have 1 X k=0 jjE e X(x)jL (k) 2 ( 0) jj 2 a2 k = X  1 A0 ] <1 i.e.,E e X(x) is a member ofXe(H). It is easy to show Z 0 !(y)E e X(x;y)dy= 1 jHj X  a! ] A0 ] e;i2H ;1x 

with a! ] the Fourier coecient of !(x). Since !(x) 2 L

2(0), it

follows that the setfa! ]j 2Zngis bounded. Hence, the

convolu-tion (E e

X!)(x) is an innitely dierentiable function which satises

(1.2). The norm square of the convolution may be written as

1 X k=0 jjE e X !(x)jL (k) 2 ( 0) jj 2 a2 k = 1jHj X  ja! ]j 2 A0 ] <+1:

Thus, the convolution (E e

X !)(x) and the function ul(x) which is

dened by (2.4) are the members of Xe(H).

Now we consider the following series 0(y) = P

 (y; ), where

(y) is the conventional Dirac delta function. It is evident that0(y) is

a distribution. Moreover,0(y) is a member of S

0, the dual of the

well-known Schwartz space S. For 8 2Zn we have 

0(y+ ) = 0(y).

The Fourier transform of0(y) is equal to 0(y), i.e., e

0() =0(y)

(see, e.g., Sobolev (1974)). Whence and from the denition of the Fourier transform, we expand0(y) in the Fourier series as follows

0(y) = Z 0()e i2 yd= X  Z (; )ei 2 yd = X  e i2y:

The partial sums of this Fourier series converge to0(y) in S

0.

Let T = (11:::1) be a n-tuple. Following Vladimirov (1979, p. 120), we consider the set

D0 T =ff(x)2D 0 jf(x+ ) = f(x) 8 2Z n g:

We can dene the special inner product (f)T such that the

gener-alized Parseval identity holds (2:5) (f)T =X  cTf ]cT ] 8f 2D 0 T 82C 1 \D 0 T

where c_Tf ] = (fe;i2y)

T and cT ] = (e;i2y)

T, (see Vladimi-rov (1979, p. 121{124)). If f(y) =0(y) 2D 0 T, then (2:6) (0(y)(y))T=(0) and therefore cT 0 ] = 1

(see Vladimirov (1979, p. 121)). Whence and from (2.5) it follows that

(0(y)(y))T= X

 cT ]:

Considering the equality

cT ] = (e;i2y) T = Z Q (y)e ;i2ydy (y) 2C 1 \D 0 T

(see Vladimirov(1979, p. 122)),and performing the change of variable

y=H;1x '(x) =(H;1x), we have (2:7) (0(H ;1x)'(x)) T =X  cT' ] =X  1 jHj Z 0 '(x)ei2H ;1 x dx: By (2.6), the equality (0(H ;1x)'(x))

T='(0) holds. Whence and

from (2.7) we infer

(2:8) '(0) =X

 cT' ]:

The function (y) = '(Hy) belongs to C1 \D

0

T for an arbitrary

'(x)2 e

X(H). Consequently, (2.8) holds for 8'(x)2 e

X(H). From the denition of E

e X(x) it follows that (2:9) (E e Xei2H ;1 x )Xe = 1 6 = 0:

The Fourier series of a function'2 e

X(H) converges to'in the norm of Xe(H). Whence and from (2.9), we have

(E e X')Xe = X  cT' ]( E e Xe;i2H ;1 x )Xe = X  cT' ]:

Comparing this equality with (2.8), we see that (2:10) ((x)'(x)) = (E e X')Xe 8'2 e X(H):

Hence, the equalities (E e X(x;y)'(x)) e X = (E e X(z)'(z+y))Xe ='(y) (E e X(x;xk)'(x)) e X ='(xk)

hold and we have (ul')Xe = Z 0 !(y)(E e X(x;y)'(x)) e Xdy; X xk 2 0 ck(E e X(x;xk)'(x)) e X =Z 0 !(y)'(y)dy; X xk 2 0 ck'(xk) = (l') '2 e X(H):

With this equality established, (2.3) is immediate. ut

By the Riesz theorem every bounded linear functionall(x) on the Hilbert spaceXe(H) may be written as the inner product

(l') = ('l)Xe

(H) ' 2

e

X(H)

with l(x) the uniquely determined member of Xe(H). In terms of

Sobolev and Vaskevich (1996, 1997),l(x) is calledthe extremal

func-tion of l(x) or Xe(H)-extremal function. Since (2.3) holds, it follows

thatl(x) =ul(x). Moreover,l is a solution to the following partial

dierential equation (2:11) 1 X k=0 (;1)k a2 k kl(x) =l(x)

whereis the Laplace operator, and the series converges in the norm ofXe (H). If Xe(H) =Le

(m)

2 (H), then this equation is the

polyharmo-nic equation forl(x).

By (2.3), we can write the Xe (H)-norm square of an errorl(x) of

the form (2.1) as follows (2:12) jjlj e X (H)jj 2= (lu l) = Z 0 !(x)ul(x)dx; X xk 2 0 ckul(xk) =Z 0 Z 0 !(x)!(y)E e X(x;y)dxdy;2 X xk 2 0 ck Z 0 !(x)E e X(x;xk)dx + X xk 2 0 X xk 02 0 ckck0 E e X(xk;xk 0) = (c):

Proof of Theorem 2. Letc= (c1:::cN) be a vector of the weights.

We consider the auxiliary function

1(c) = X xk 2 0 X xk 0 2 0 ckck0 E e X(xk ;xk 0)

and check that1(c)>0 for every nonzero vectorc 2Rn.

If c 6= 0, then the linear combination c(x) =

N

P

k=1

ck(x;xk) is

a nonzero member of Xe (H). Hence, there exists a functionu

c(x) in e X(H) such that (c(x)') = ('uc(x))Xe 8'2 e X:

Whence and from (2.10) we inferuc(x) = PN

k=1

ckE e

X(x;xk). Moreover,

it is easy to show that

jjc(x)j e X (H)jj 2= ( c(x)uc(x)) = X xk 2 0 X xk 0 2 0 ckck0 E e X(xk ;xk 0)  1(c)

i.e.,1(c) is a strictly positive function.

Since (c) 0 and 

1(c) > 0 for c

6

= 0, it follows that (c) is an unbounded function in a neighborhood about innity. Moreover,

(c) attains a minimum inc=c0

2Rn. A rule of the form (2.1) with

the weightsc=c0 is the e

X(H)-optimal cubature formula.

We apply the strict convexity property of the unit ball in Xe(H)

to prove a uniqueness of the optimal error in Xe (H). We recall the

strict convexity property.

Let l1l2:::lk be some functionals belonging to the unit sphere

of Xe (H) and let 

12:::k be nonnegative reals with sum 1.

Then the respective linear combination of lj lies in the unit ball of

e X (H), i.e., k k X j=1 jlj j e X (H)k1:

Equality holds here if and only if alllj coincide.

From strict convexity it is immediate that there is a unique op-timal error in Xe (H). For, were it otherwise, we would nd at least

two errors with the same minimal norm. As follows from the last in-equality, their half-sum would then have the norm less than each of them a contradiction. ut

3. The dierence operators of Sobolev type

In this section, we introduce the dierence operators of the special form. These operators will be used to construct desired weights of

e

X(H){optimal cubature formula explicitly. Consider the auxiliary function;(

e X) hH (p) dened by ;( e X) hH (p) X  1 A0H p ;h ;1 ] = ( ;1)m p2Rn: The function ;( e X)

hH (p) is periodic in p with period matrixh;1H ;1,

;( e X) hH (p+h;1H ;1 ) =; ( e X) hH (p) 8 2Zn: We can expand ;( e X)

hH (p) in the Fourier series

;( e X) hH (p) = X  D ( e X) hH ]e;i2phH where D( e X) hH ] = 1j 1 j Z 1 ;( e X) hH (p)ei2hHpdp 2Zn:

Here the integral spreads over the fundamental parallelepiped 1 of

the matrixh;1H ;1.

Now we dene the convolution

D( e X) hH ]' ] X  D ( e X) hH ;]']

for a compactly-supported function ' ]. The convolution D( e

X)

hH ]

is an analog of the partial dierential operator in the left side of (2.12). It is usual to use compactly-supported analogs of the linear dierential operator with partial derivatives in the theory of nite dif-ferences. The convolutionD(

e

X)

hH ]is constructed in another fashion

and not compactly-supported. If Xe(H) = Le (m)

2 (H), then the

con-volution D( e

X)

hH ] coincides with the operator D (m)

hH ] which was

suggested by S. L. Sobolev as a dierence analog for the polyhar-monic operatorm _{(see Sobolev (1974)). The properties ofD}(

e

X)

hH ]

and D(m)

hH ] are similar in many aspects. By this, we callD (

e

X)

hH ]

the dierence operator of Sobolev type. We have the following

Lemma 2.

;( e

X)

hH (p) is a real and analytic function for all p 2 Rn.

In a neighborhood about zero ;(m)

hH (p) may be written as (3:1) ;( e X) hH (p) = (;1)mA 0H p]  1 + X jj>0 b !(hp)  : The function D( e X)

hH ]decreases with j j growing. There are positive

constants C and s such that for all the inequality holds

(3:2) jD ( e X) hH ]jCj j 1=2e;sjj:

There is a similar lemma in Sobolev and Vaskevich 12, p. 340]. For the proof in full detail, we refer to Vaskevich 13].

From Lemma 2 it follows that the convolution

D( e X) hH ]' ] X  D ( e X) hH ;]']

exists for a function' ] increasing withj jgrowing as degree ofj j.

From the denitions of ;( e X) hH (p) and D( e X) hH ] it follows that ;( e X) hH (;p) =; ( e X) hH (p) D( e X) hH; ] =D ( e X) hH ]:

Theorem 3.

If1=h is a positive integer, then the functionE (h) e

X  ] =

E e

X(hH ) of a discrete variable and the operator D( e X) hH ] are re-lated by (3:9) (;1)mhnj 0 jD ( e X) hH ]E (h) e X  ] =h ]:

Here h ]equals 1 in the points of setf=hj2Zngand 0 outside

of the set. If ] = 0for 6= 0and 0] = 1, then

h ] =X

  ;=h]:

Proof. By the denition of E e X(x), we have (3:10) X  E (h) e X ]e;i2hH p= 1 jHj X  1 A0 ] X  e ;i2h(H p+):

Substituting the equality

X  e ;i2h(H p+) = 0(hH p+h ) = X  (hH p+h ;)

into (3.10), we interchange the order of summation over and. So, the sum in the left side of (3.10) equals

1 jHj X  1 A0H p ;h ;1] X  (hH p+h ;):

Given a positive integer 1=h, it is easy to show

X  (hH p+h ;) = X 0 (hH p;h 0):

Consequently, the product;( e X) hH (p)P  E (h) e X ]e;i2hH pequals (3:11) 1 jHj ;( e X) hH (p)  X  1 A0H p ;h ;1]  X  (hH p ;h )  : By the denition of;( e X) hH (p), we have (3:12) ;( e X) hH (p) X  1 A0H p ;h ;1 ] = ( ;1)m p2Rn:

Whence and from (3.11), we infer (3:13) ;( e X) hH (p) X  E (h) e X ]e;i2hH p = (;1)m jHj X  (hH p ;h ) = ( ;1)m hnjHj X  e ;i2phH(=h):

The Fourier coecients of the product of two functions result from convoluting the Fourier coecients of the factors. Considering this and substituting the expansion

;( e X) hH (p) = X  D ( e X) hH ]e;i2phH in (3.13), we come to (3.9). ut

4. The weights of the

f

X

(

H

)-optimal cubature formulas

In this section, we consider the cubature formulas of the form (4:1) (l') =Z 0 !(x)'(x)dx; X hH2 0 c ]'(hH )

where!(x) is the weight function that was introduced in section 2.

Theorem 4.

The Xe(H)-optimal cubature formula of the form (4.1)

has the following weights

(4:2) c(!) 0  ] =h n₍;1)mjHj Z 0 !(x)D( e X) hH ]E e X(x;hH )dx where hH is a vector in 0. Proof. Let c = fc ] j hH 2  0

g be the vector such that the

quadratic form(c) dened by (2.12) attains a minimum in c. The vectorc is a unique solution to

(4:3) X hH0 2 0 c 0] E e X(hH ;hH 0) =f ] hH 2 0 where f ] =Z 0 !(x)E e X(x;hH )dx: If c ] = 0 for hH =2 

0, then we can change the sum over 0 in

(4.3) by the sum over 0

2Zn. Considering this, we write (4.3) can

be written in the convolution form

(4:4) E (h) e X  ]c ] =f ] h 2Q whereE (h) e X  ]E e X(hH ). Letv ] =E (h) e X  ]c ]. By the periodicity of E e

X(x), we infer v +h;1] = v ] and f +h;1] = f ] for 8 2Zn. Hence, v ] andf ] are completely dened by the values fv ]jh 2Qgand ff ]jh 2Qgrespectively. Whence and from

(4.4) it follows that

(4:5) E

(h) e

X  ]c ] =f ] 8 2Zn:

Now we apply the convolution operatorD( e

X)

hH ]to both the sides of

(4.5) and obtain (D( e X) hH ]E (h) e X  ])c ] =D ( e X) hH ]f ] 8 2Zn:

Inserting (3.9) into the left part of this equality, we infer c(!) 0  ] c ] = X  c ;h ;1] =h ]c ] = (;1)mj 0 jhnD ( e X) hH ]f ]:

With this equality established, (4.2) is immediate. ut

Corollary 1.

The weightsc(!)

0  ]of the e

X(H)-optimal cubature for-mula of the form (4.1) may be written as follows

(4:6) c(!) 0  ] =h nX  (;1)m; ( e X) hH (H ;1) A0] e;i2ha !]

with hH a vector in 0 and a!] the Fourier coecient of the

weight function !(x).

Proof. The weight function !(x) which was introduced in section 2 is a locally integrable function. Whence and from the periodicity of

;( e

X)

hH (H ;1), it follows that the series in (4.6) converges absolutely

and uniformly. Applying to (2.2) the convolution operatorD( e X) hH ], we have (4:7) D( e X) hH ]E e X(x;hH ) = 1 jHj X  1 A0] D( e X) hH ]e ;i2H ;1 (x;hH): By the denitions ofD( e X) hH ] and; ( e X) hH (p), we infer D( e X) hH ]e ;i2H ;1 (x;hH) =e;i2H ;1 (x;hH) X  D ( e X) hH]e;i2H ;1 hH =e;i2H ;1 (x;hH); ( e X) hH (H ;1):

Whence and from (4.7) it follows that the convolution D( e X) hH ] E e X(x;hH ) equals (4:8) 1 jHj X  ;( e X) hH (H ;1) A0] e;i2H ;1 (x;hH):

Substituting (4.8) into (4.2), we interchange the integration over x

and the summation over. By choosing ; as a new variable of the

5. The error of the

f

X

(

H

)-optimal cubature formulas

Proof of Theorem 1. Let !(x) = ei2H ;1 x

. By (4.6), we can write the weights of theXe(H)-optimal cubature formula of the form (4.1)

as follows c() 0  ] =h njHj (;1)m; ( e X) hH (H ;1) A0] ei2h:

With this equality established, (1.5) is immediate.

We estimate the error of the cubature formula with the weights

c()

0  ] on every space of the family f

e

X (H)j(ak)g.

Given an error of the form (4.1), from (2.12) we have

(5:1) jjlj e X jj 2= (lu l) =jjulj e Xjj 2 = jHj X  A0] jau l] j 2:

The Fourier coecientaul] of the extremal function ul(x) depends

on L]  (l(x)ei 2H ;1 x ). By (2.3) with '(x) = ei2H ;1 x , we have L] = (ul(x)e;i2H ;1x )Xe = 1 X k=0 j2 H ;1 j 2k a2 k Z 0 ul(x)ei2H ;1x dx= 1 jHj A0]au l]:

Apply rst this equality and next (5.1). Then we infer (5:2) jjlj e X jj 2= 1 jHj X  jL]j 2 A0] :

Based on (5.2), we nd the Xe (H)-norm of the Xe(H)-optimal error

explicitly. Let us compute the Fourier coecients L],  2 Zn, of

the optimal error.

Given the weights c(!)

0  ], we denote by L (!)

0 ] the respective

Fourier coecientsL]. If!(x) =ei2H ;1 x

, then we agree to write

L()

0 ] instead of L (!)

0 ]. Apply rst the denition of L (!)

0 ] and

next (4.2). Then, by (5.1), we infer

L(!) 0 ] =a! ;]; X hH2 0 c(!) 0  ]e ;i2H ;1 hH=a !;] ; Z 0 !(x)h hnX h2Q ei2h(;) i X  (;1)m; ( e X) hH (H ;1) A0] e;i2H ;1x dx:

Let (;)h be a vector with the integer entries. Then h hnX h2Q ei2h(;) i = 1:

Otherwise, the value in brackets equals zero. Hence

L(!) 0 ] =a! ;]; X  (;1)m; ( e X) hH (H ;1(+=h)) A0+=h] a!;;=h]: Let!(x) =ei2H ;1  x. Then (5:3) L() 0 ] = jHj h 1; (;1)m; ( e X) hH (H ;1) A0] i  L() 0 +=h] = ;jHj (;1)m; ( e X) hH (H ;1) A0]  6= 02Zn:

If there exists a noninteger entry of a vector h(;), then L () 0 ]

equals zero.

Given anXe-optimal cubature formula of the form (4.1), we intend

to nd the Xf 1 -norm of the e X-optimal error l() 0 (x). Here f X1 is a

Hilbert space from the family under consideration. Let Ae

0 ] be the

function (1.4) in case of the spaceXf

1. From (5.2) and (5.3) it follows

thatjjl () 0 j f X1 jj

2 equals the product of jHj " A2 0] e A0]  X 6=0 1 A0 ; =h]  2 +X 6=0 1 e A0 ; =h] # and " 1 +X 6=0 A0] A0 ; =h] # ;2 : IfXf 1= e X, then we have (1.6). ut

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