Investigation of the optimization of quadrature formulas by Dnepropetrovsk mathematicians

54. D. Jackson, "0ber die Genauigkeit des Anniherung stetiger Funktionen durch ganze ge- gebener Ordnung," Dissertation, G6ttingen (1911).

55. N. P. Korneichuk, "Sharp constant in Jackson's theorem on best uniform approximation of continuous periodic functions," Dokl. Akad. Nauk SSSR, 145, No. 3, 514-515 (1962).

56. N. P. Korneichuk, "Sharp constant in Jackson's inequalities for continuous periodic functions," Mat. Zametki, 32, No. 5, 669-674 (1982).

57. V. V. Shalaev, "Approximation of continuous functions by trigonometric polynomials," in: Studies on Current Problems of Summation and Approximation of Functions and Their Application [in Russian], Dnepropetr. Univ., Dnepropetrovsk (1977), pp. 39-43.

58. N. I. Chernykh, "Best approximation of periodic functions by trigonometric polynomials in L2," Mat. Zametki, ~, No. 5, 513-522 (1967).

59~ A. A. Ligun, "Sharp constants of approximation of differentiable periodic functions," Mat. Zametki, 14, No. i, 21-30 (1973).

60. A. A~ Ligun, "Sharp constants in Jackson type inequalities," Dokl. Akad. Nauk SSSR, 28__/3, No. I, 34-38 (1985).

�9 ," Dokl. Akad. Nauk 61 V.A. Yudin, "Diophantine approximations in extremal problems of L 2 SSSR, 25__ ! , No. I, 54-57 (1980).

62. A. A. Ligun, "Sharp inequalities of Jackson type for periodic functions in the space L2," Mat. Zametki, 43, No. 7, 757-768 (1988).

INVESTIGATION OF THE OPTIMIZATION OF QUADRATURE FORMULAS BY

DNEPROPETROVSK MATHEMATICIANS

V. P. Motornyi UDC 517.5

A survey of results is given on the extremal problems of the theory of quadratures, obtained by mathematicians whose scientific activity has been connected with the Dnepropetrovsk University and, in particular, with the work of the scientific seminar, conducted for many years by N. P. Korneichuk.

I. Introduction. The development of numerical analysis methods has led to the following extremal problem in quadrature theory. In the space C r-1 of functions, r - i times continuously differentiable on the segment [0, I], there is given some class H. A definite integral of the function [ 6 E is evaluated with the aid of the quadrature sum

L (f; x , P) = V V p j ' (xh), ( i )

where O<~p<~r--l, 0~x,<x2< ...<xm<~- I, pware arbitrary. The error R(f; X, P) of the computation of the integral is defined by the quadrature formula

[(x)dx = ~ + Ph.f" (x~) + Rq; X,P), (2) 0 k ~ l l~O

where X = {Xk} is the vector of the nodes, P = {PkZ} is the vector of the coefficients, f(~ = f(x). By A we denote the set of the pairs of vectors (X, P), which may be arbitrary or may satisfy certain conditions, defining the properties of the quadrature formula (2). The quantity

R (//, X, p) = sup I R (f; X, p)[ fe l t

defines the largest error of the quadrature formula (2) on the functions of the class H. One has to find the quantity

8~ (H) = im R (/4; X, P), fX, P)EA

Dnepropetrovsk University. Translated from Ukrainskii Matematicheskii Zhurnal, Vol~ 42, No. i, pp. 18-33, January, 1990. Original article submitted August 22, 1989.

0041-5995/90/4201-0013512.50 �9 1990 Plenum Publishing Corporation 13

and also to indicate the vectors of the nodes X* = {x~} and of the coefficients P* = {p~} (if they exist) for which the infimum is attained. The quadrature formula (2) with nodes x~ and coefficients p~ is said to be optimal or the best among all such quadrature formulas on the class H.

The formulation of this problem and the first basic results belong to Nikol'skii [i-3], whose monograph "Quadrature formulas" has undoubtedly increased the interest in the extremal problems of the theory of quadratures. Nikol'skii [i] has suggested a method which reduces the solution of the extremal problem of the theory of quadratures to the minimization of the norm, determined by the selection of the class H, of some piecewise polynomial function, i.e., monospline, as it is called now, of degree r and defect p + 2. Most likely, for small r and P = 0, this method has been known to specialists in numerical mathematics (see, for example, [4]).

The actuality, the complexity, the connection of the extremal problems of the theory of quadratures with the constant problems of the theory of approximations, manifesting itself in the nature of the investigation methods, with the theory of spline functions, have at- tracted the interest of N. P. Korneichuk, who, starting from the sixties, has stimulated the development of this area at the Dnepropetrovsk school of approximation theory. We mention that the founder of this school is S. M. Nikol'skii.

To N. P. Korneichuk, to his students and colleagues - researchers at Dnepropetrovsk University - belong a series of important result in the optimization theory of quadrature formulas. The purpose of this paper is to give a survey of these investigations. We present some fundamental results obtained by the participants of the scientific seminar of the Chair of Theory of Functions at the Dnepropetrovsk University. N. P. Korneichuk conducted this seminar between 1963 and 1974 and up to now he has maintained a close relation with the Chair of Theory of Functions at the Dnepropetrovsk University, continuing to exercise in- fluence on the topics and the efficiency of the seminar.

2. The Beginning of the Investigations. We introduce some definitions. By WrLp we denote the class of functions, defined on the segment [0, i], having absolutely continuous derivative of order r - I, and the r-th derivative of which satisfies the condition

I

~ ][~'~(x) JPdx<~l for l-~<p<~ , while for p = ~ almost everywhere l/(r~(x)[~<l. The corresponding

0

classes of periodic functions will be denoted by WrLp. We shall consider the cases when the period is 1 or 2~, specifying this in each concrete case.

Let O<~x1<x=...<x m ~<I. By a monospline of degree r, defect p + 2, and order m we mean a function Gr(t), defined on the entire axis, having continuous derivative of order r - 9 - 2 if O < r - 2, continuous if O = r - 2, and coinciding on the intervals (--co, xx), (x~, xk~.0, k = I, 2 ...... m-- I, (xm, oo)with polynomials of degree r, for which the coefficient of t r is equal to i. The points x k are called the nodes of the monospline. By ~, we denote the set of monosplines which on the intervals (-~, xl), (x m, ~) are equal to t r and (t - i) r, respectively. Each monospline OrE~ can be represented in the form

m p

EE r, O,(t) = ( t - 1) - - ( - - 1 ) " (r-- l--I)I Ph~K,--~(xh --0. ( 3 )

k ~ l l = 0

where Kv(t) = t v-1 if t~0, and Kv(t) = 0 if l~0, while the coefficients Pk~ of the monospline are such that Gr(t) = t r for t < x I.

By ~r we denote the set of 1-periodic monosplines with nodes at the points 0 = x I < x 2 < ... < x m < I, which belong to the class cr-P -2 if p~r--2 and on the intervals (x k, Xk+ l) coincide with algebraic polynomials of degree r, whose coefficient of t r is equal to i. Each

--O

monospline G~E~, can be represented in the form

nl o

G, (t)= ~ ~ ( - l)'r! * t , . : ~ (r - - l)! p~zB~_., ( - - xk).

where B~(t) are the Bernoulli polynomials (see, for example, [4]), while ~P~0----I.

It i s n o t a c c i d e n t a l t h a t t h e n o d e s and t h e c o e f f i c i e n t s o f t h e q u a d r a t u r e sum ( 1 ) and the nodes and the coefficients of the monosplines in the representations (3) and (4) are

(4)

14

denoted by the same symbols. S. M. Nikol'skii has established that the error of the quadra-

ture formula (2), exact on polynomials of degree r - i (resp., exact on constants), on the functions of the class WrLp_ (resp., of the class WrLp), is expressed in terms of G:6~

(resp., in terms of GrE~r ) by the formula

R(/;x,P) ~--1)' = r j G( t ) / '~ ( t )d t , (5 ) 0

and, m o r e o v e r , t h e nodes and t h e c o e f f i c i e n t s o f t h e m o n o s p l i n e s G r c o i n c i d e w i t h t h e nodes with the coefficients of the quadrature sum (i) and

I R(W'Lp; X,P) = 7NIlGIIL q, G,E~%,, (6)

where p and q a r e m u t u a l l y c o n j u g a t e . In t h e p e r i o d i c c a s e we have

1 R(~V'Lp; X,P) = -/TIIGllL: GE3~. (7)

on t h e o t h e r hand , t a k i n g any m o n o s p l i n e G r f rom ~ w i t h nodes x k and c o e f f i c i e n t s Pk~ and constructing the quadrature formula (2) with the vector of nodes X = {Xk} and vector of coefficients P = {PkZ}, we obtain [3] a quadrature formula, exact for polynomials of degree r - i, whose error for any function fEWrLp is expressed by formula (5), while the error on

the class WrLp by the formula (6). Similarly taking any monosplineG~r: -~ c with nodes x k and coefficients PkZ and constructing the quadrature formula (2) with the vector of nodes x = {Xk} and vector of coefficients P = {PkZ}, we obtain a quadrature formula, exact on constants, whose error on any function fE -~ . , Wip is expressed by formula (5) while the error on the class Wrip by formula (7).

From equality (6) and from the one-to-one correspondence between the set of quadrature formulas of the form (2), exact on polynomials of degree r - I, and the set of monosplines ~ there follows the equality

1 ~(W'Lv)= inf ~ l lG~! rL , . (8 )

Correspondingly, in the periodic case we obtain the equality

9 " ~r , g,,~(~ kp)= inf ~ [ I G [ t ~ . (9)

The reduction of the problem of the determination of the best quadrature formula, expressed by the relations (8), (9), has promoted in the same degree the investigation of spline functions and the solution of the extremal problems of the theory of quadratures.

The first results at the Dnepropetrovsk University, regarding the optimization of quadrature formulas, have been obtained by N. P. Korneichuk's student N. E. Lushpai [5]. He has found the best quadrature formula of the form (2) for p = r - i, r = i, 2 .... , on the classes WrLp and W~L~ ={rE W'L~, fk(0) =0, k =0, i ...... r--I} and for p = r - 2, r = 2, 4 ..... on the ciasses-W~Lp. -

The problem of the best quadrature formula of the form (2) for the classes W{Lp for p = 0 and r = I, 2, and also for p = r - 2, r even, has been solved by Nikol'skii [i], while for p = 1 and A. I. Kiselev (see [3], Sec. 14). Simultaneously with N. E. Lushpai and independently, the general case of this problem I~ p<oo for 0 = r - i, r = i, 2 ..... and

p = r - 2, r = 2, 4 ..... has been considered in [6, 7]. In the periodic case, N. E. Lushpai has obtained the following result.

THEOREM i [18]. Among all quadrature formulas (2) for p = r - i, r = I, 2 ..... and -t

p = r -- 2, r = 2, 4 ..... the best for the class WLp, l~.~p~oo, is the formula

1 m [ T ~

. = \ /+ 0 k=~ 1=0

where Rrq(t ) is an algebraic polynomial of degree r, with the coefficient of t r equal to i, least deviating from zero in the space Lq. In this case,

15

g~ (~r~ p) = R~, (1)/r! ~ rq + i (2m) ~.

However, as the first essential success one can consider the results obtained by Kor- neichuk and Lushpai [9]. In that paper one has solved the problem on the best quadrature

formula of the form (2) for p = r - 2 and p = r - 3 for the classes WrLI, W~L z, and WrL z for r = 3, 5,..., one has proved the uniqueness of the optimal quadrature formula (in the periodic case the uniqueness to within a shift of all the nodes by the same amount), one has obtained a characterization of the monosplines from the sets ~[~ ~ and fi~ , least deviating from zero

in the space C (see also [3], Theorems E.2, E.3, E.6, E.7), and one has focused attention on the question of the existence of an optimal quadrature formula. The investigation has re- quired fundamental new methods for the determination of monosplines, least deviating from zero in the space C, since in the considered case (unlike those mentioned above) the monosplines are continuously differentiable. One has succeeded to apply these methods [i0, Ii] also to

the class WrL 2.

THEOREM 2. The best quadrature formula of the form (2) on the class WrL 2 for p = r - 2 and P = r - 3, r = 3, 5 ..... has the form

t m ~ r - - 3 ) / 2

I I j ' o . - . ~ 7 I 2 .~.-2~-',, i f12. ,, "~ - - 1 ~ + R (f),

[(x) dx = -_.2_r ! ,.~ "~', _.# ' (2m) 2'~-1" Ll~ ~-2l-l~ (1)-- ( 2 r - - 1 ) ( r - - 1) ~- t (1) i m 1 ~1 fi=l i = 0

where s is the Legendre polynomial of degree r. In this case,

f (x~ d x -

0

In this case

r-

- :" J , ' ( r - - 1! (r --~ 2~ 1 8~i, fWrL,) == [2r) ! t r t , r-- 1)(2r .+ 1) tn ~

We should mention also the results of V. M. Alkhimova, who has considered for the classes WrLp the Markov type quadrature formulas

1 r - - I m - - I p

l'[(x) cix = '~, (po,f u) (0)4- pmzf (') (1)) + Z V p , t l " ) ( x , ) + R (f), ( lO) 0 l = 0 h ~ l / = 0

where 0 < x z < x 2 <... < Xm_ z < i.

Making use of S. M. Nikol'skii's scheme of arguments, Alkhimova has obtained the following result [12], obtained independently also in [13].

THEOREM 3. Among all quadrature formulas of the form (i0) for p = r - i, r = i, 2, .... and p = r - 2, r even, the best on the class WrLp for all p is the unique formula

l ~-, .__~m-'I~/2] t~'r-iz--"(1) / k ) .~ . ~ L,/#~ (0) + f":' (I)i + X -~ X "~ " '~ ' r~2t~

r! :_: (2tn) ~-I ~ , ~ (2m) 2z+l ' i -'~" + R (f). i ~O ktml l ~ 0

sup R(h = R.q (1) ~W~Lv 2rr I ~:rq + 1 m"

V. M. Alkhimova has considered also other quadrature formulas with fixed nodes at the endpoints of the interval for the classes WrLI and WrLi. Quadrature formulas of Markov type

have been investigated also in [14, 15].

3. The Optimization of Quadrature Formulas by the Estimation of the Remainder on Func- tions Reducing the Quadrature Sum to Zero. Let X = {Xk} be some vector of nodes. We denote

by H X the subset of functions from H such that f"~(x~) = O, k= I, 2 ..... l=O, 1 ..... p. If H can be represented in the form H = {f + cIf E H0, cE RI}, where H 0 is a convex, centrally symmetric, compact set from the linear space ~ , then for any vector of coefficients we have

Consequently,

I

R (H; X, P) = R (Hx; X, P) = sup ~ f(x) dx . I fHx 0

Many classes, considered in the theory of quadratures,

I

8~(H) = infx feMxSUp ! f(x) dx. (11 )

satisfy the above indicated conditions;

16

in particular they are satisfied by the classes WrLp, WILp and H w, WrHW, defined below. Prob- ably, in the general case, the computation of the quantity in the right-hand side of the

equality (11) is not simpler than the computation of the right-hand sides of the relations (8), (9). However, in the cases when there exists weighty arguments allowing us to assume that precisely the quadrature sum L(f; X0, P0) restores optimally the integral on the class H, one can attempt to prove that for any vector of the nodes X there exists a function fx 6 H X such that

I

S fx (t) dt > R (H, X0, P0), ( 12 ) 0

of c o u r s e , u n d e r t h e c o n d i t i o n t h a t a l s o t h e r i g h t - h a n d s i d e o f (12) can be found .

We a l s o n o t e t h a t t h e r e d u c t i o n o f t h e e x t r e m a l p r o b l e m o f t h e t h e o r y o f q u a d r a t u r e s , s u g g e s t e d by S. M. N i k o l ' s k i i , i s n o t a l w a y s a p p l i c a b l e , f o r e x a m p l e , f o r t h e c l a s s e s wrHw. Naturally, in this case it is necessary to attempt to realize the above given scheme of arguments. Such an approach has been repeatedly used for the determination of order estimates. The perspectivity of using this for obtaining sharp results has been pointed out by Korneichuk in [16], in which one has obtained optimal cubature (in particular, quadrature) formulas on classes of functions of several variables with a given majorant of the modulus of continuity. G. K. Lebed', a participant of the seminar conducted by No P. Korneichuk, has reacted at once to this. He has considered the problem of the best quadrature formula of the form

m

( q (x) [ (x) dx = \~ p j (xh, %" ,9,~ (f; X, P) ( 13 )

on the class H w, where q(x) is a nonnegative, integrable function H w ={] : I] (h)--[ (~2) ! (It*--l~[). where w(t) is some modulus of continuity. Lebed' [17] has shown that for a fixed

vector X of nodes, the optimal with respect to the coefficients on the class H w is the quadrature formula (13), whose coefficients are computed by the formulas

p~ = ~" q ',x) ax, (14) At,

where A k is the set of the points of the segment [-i, I], lying closer to the point x k than to the remaining nodes. In this case,

Rq~(H~; X)= inf sup Rq(f; X, P)= ~ I q(x)~(Jx--xh[)dx" (15) P fEh,~

k ~ l A k

C o n s e q u e n t l y , t h e d e t e r m i n a t i o n o f an o p t i m a l q u a d r a t u r e f o r m u l a o f t h e fo rm (13) r e d u c e s t o t h e m i n i m i z a t i o n o f t h e r i g h t - h a n d s i d e o f t h e r e l a t i o n (15) w i t h r e s p e c t t o t h e n o d e s , In [17, 18 ] , Lebed f h a s c o n s i d e r e d some s i m p l e c a s e s o f t h e m i n i m i z a t i o n o f t h e r i g h t - h a n d s i d e o f t h e e q u a l i t y ( 1 5 ) ,

xk-i§ , k = 2, 3,. m, and in [17] The sets A~ = (~h, ~+t), whereat =--I, ~m+~ = I, �9 = 2 "''

they have been defined in this manner. The above introduced definition admits a generalization to the case of cubature formulas. One has obtained the following generalization: Babenko [19, 20] has established analogues of the relations (14), (15) and, in addition, has con- structed asymptotically optimal cubature formulas for classes of functions of two variables with a given majorant of the modulus of continuity. For the class H a [H a = H w if w(t) = t a, 0 < a ~ i). Babenko [20] has found the asymptotic behavior of the error of a sequence of optimal cubature formulas, under the conditions that q(x) is a nonnegative, bounded, Jordan measurable function. In the case of a function of one variable this asymptotic behavior is of the form

I l

gq,~(H~)=inf sup R q ( f ; X , P ) = (2m)- '~{f q'g'4-i'(x)d + o . (16)

--1

4. O p t i m a l Q u a d r a t u r e F o r m u l a s f o r C l a s s e s o f P e r i o d i c F u n c t i o n s . Among a l l q u a d r a t u r e f o r m u l a s o f t h e fo rm ( 2 ) , o f g r e a t e s t i n t e r e s t a r e t h e q u a d r a t u r e f o r m u l a s f o r p = 0:

~ f (x) dx = Phf (x~) + R (h. ( 17 ) 0 k = l

17

This can be explained by their simplicity and by the fact that we do not have always information on the derivatives of the function f. In addition, it turns out (see Sec. 5) that the restoration (approximate evaluation) of an integral, from information determined by the values of the function, is optimal for some classes of functions in comparison with the methods that use also information on the derivatives and having the same number of units of information.

First we consider the optimization of quadrature formulas of the form (17) for classes of periodic functions, by using the general considerations of Sec. 3. Moreover, it is convenient to consider functions of period 2~ and then formula (17) assumes the form

2n m

f (x) & = a / ( x O + R if). ( 18 ) 0 ~i

Returning to the inequality (12), we note that the quantities R(H; X0, P0) can be easily computed for the classes of 2v-periodic functions if the vector X 0 of the nodes is defined by the points x k = 2kz/m, k = 0, l,...,m- i, while the vector P0 of the coefficients by the numbers Pk = 2~/m and H is a convex, centrally symmetric set, containing with each function f(x) also the functions f(x) + c and f(x + a), where c and a are arbitrary constants. The quadrature formula corresponding to the indicated vectors X 0 and P0 has the form

2:1 ,;t--i

S . i 2krt --V , , , / + R(I) (19) U :r

and it is called the rectangle formula.

In this case [21] we have

R (H: X,, , Po) = 2 ~ , ~ ( H ) ,

where ~=(H) = sup max lf(x)l, while H m, m = i, 2, .... is the set of functions from H of period f 6 t t m x

2~/m with zero mean value over the period.

The quantities Um(H) and the functions realizing this supremum have been computed for various classes H in function theory investigations, for example in Bernshtein's paper [22], in S. B. Stechkin's appendix to the Russian edition of [23], and in Korneichuk's paper [24]. Thus [22],

tt,, (Ts = ~ , -. , r = 1 9 , ( 2 0 ) lT~r ' . . . . ,

where ~r = ~rL~ and K r are the Favard constants.

Let WrHm be the class of 2~-periodic functions for which !/r~ (if)__ f(rl (l~)] ~ ~ (Itl--t~ I) , where m(t) is a given modulus of continuity.

For the class WrHm we have the equality [24]

�9 - F r r ~ ~ ( W , v ) = m a x l / ~ r ( t ) ] , ( 2 1 ) t

where ~(t) is a convex modulus of continuity, while fmr(t) is a function from the class WrH~ with period 2~/m and mean value on the period equal to zero, for which the r-th derivative is odd and is defined by the equalities

f ( r ) m r ( t ) -----

J% i _ _ o~ (2t), 0 ~< t ~ 2m ;

2

ea ( 2 ~ / m - - 2t), 2.,t--T ~ t ~ ,--7

From S. B. Stechkin's inequality [23, p. 385 of the Russian edition] there follows the equality

,,?,, . . . . ( 2 2 ) p~. (,~ L = / ( r_ t / (4 r t z ), where ~s ----- l ~ r L i , r ----- 2, 4 . . . . .

From the relations (19)-(22) we obtain the equalities

~ = , 2 , , ( 2 3 ) R ( ~ ' ; Xo. Po) = - ~ K , / m , r 1 . . .

18

R ( ~ ' H ~ ; Xo, Po) = 2a max I/mr(l)1, r = I, 3 . . . . . ( 2 4 )

R (W'L; X o, Po) = ~Kr-J(2mr), r = 2, 4 . . . . . ( 2 5 )

We mention that the equalities (23), (24) have been obtained by straightforward calculations by Malozemov [25, 26].

The optimality of the rectangle formula among all formulas of the form (18) on the classes ~r for r = i, 2 follows from Theorem i. However, this case is simple and could have been known earlier. In order to establish an analogous fact for r = 3, T. N. Busarova has overcome significant difficulties and has carried out sufficiently complex constructions. In the general case the optimality of the rectangular formula among all quadrature formulas (18) follows from the inequality (12), equality (23), and from the following theorem [21, 28].

THEOREM 4. Assume that for r= I, 2 .... ~t~ there exists a set of functions f ~:'~, whose

derivatives f~r) changes sign 2m times on the period, taking alternately the values 1 or -I. x)

Then for any system of points X {0 x.,<&< x ..... ~2~} there exists a function t.,~ '~

such that minfx(t) = fX(Xk) = 0, k = 0, 1 ..... m - I, and

2~

2 a K . ( 2 6 ) ffx(t) dt~> ~" O 0

The method of the proof of the existence of the function fx(t) allows us to establish the existence of monosplines and perfect splines, vanishing at a given system of points. More- over, this method works also in the case when the method of proof, based on the application of Borsuk's theorem, is not applicable. The proof of the inequality (26) has been carried out with the use of the technique of E-rearrangements, developed by N. P. Korneichuk. Simulta- neously one has obtained estimates for the norms of perfect splines, which have application in the determination of widths and in the problems of the optimal restoration of functions and functionals~

For the class WrHm the optimality of the rectangle formula among all quadrature formulas of the form (18) has been obtained for r = i, 2 .... in [21, 28] and for r = 2 in [29]. The proof is based on the use of the equality (23) and on an assertion similar to Theorem 4. In [21, 28] one has found the best quadrature formula of the form (18) for the classes WrL (r is even) and one has also established that for the classes ~r, r > I, WrH~, r = i, 3, .... WrL (r is even) among all quadrature formulas of the form

S = ' " ' tri (2:) f (X) dx Pal ~x:) + pkf' (xl,) w

the rectangle formula is optimal.

As a consequence of the given results we obtain the solution of the problem on mono-

splines from the sets '~m, and ~m,. the last deviating from zero in L l for all r and in C for all even r'.

Solving in a straightforward manner the problem of the minimization of the corresponding 20 norms of monosplines from the set o~mr, the optimality of the rectangle formula among all

quadrature formulas of the form (18) and (27) for the class WrL (r is even) has been proved by Ligun [30], while for the class WrLp, 1 < p < ~, by Zhensykbaev [31-34], a student of N. P. Korneichuk.

In [31 32] Zhensykbaev has obtained characteristic properties of the monosplines from U0 ~,, the least deviating from zero in Lq (p and q are conjugate) and has proved its uniqueness under the condition of its existence [i.e., under the condition that the best quadrature formula of the form (18) for the class WrL~, I < p < ~ exists] The existence of a mono-

spllne from ~m,, the least deviating from zero in Lq, has been proved in [33, 34].

In 1981, the problem of the best quadrature formula of the form (2) in the general form

for the classes ~Lp, 1 ~p~oo, has been solved by the Bulgarian mathematician Bojanov [35]. It should be also mentxoned that in earlier investigations, Lushpai [36, 37] and Zhensykbaev [38] have obtained a series of interesting results regarding the solution of this problem.

19

5. The Optimization of Quadrature Formulas for the Classes WrLp. On Optimal Methods

for the Restoration of an Integral. In the nonperiodic case. the existence of optimal quadrature formulas of the form (17) for the classes WrLp, !-~<p~ ~, has been established by Bojanov [39, 40], while the uniqueness for l<p~ ~ by Zhensykbaev [41, 42]. In [41, 42], Zhensykbaev has obtained characteristic properties of the vector of the nodes X = {Xk} and of the vector of the coefficients P = {Pk} of the optimal quadrature formula of the form (17), consisting of the following.

We consider the system of equations

i r--! I Gr (t) [q-lK r (x h - - t) Sgll G r (l) dt + E s = O, k = 1 . . . . m,

o .-o ( 2 8 ) 1 r - - I

V n n--! P h t I Or (t) I~ (xh - - t) sgn Or (t) dt + z_2 ~ - - 1 )~x,~ = O, k = i . . . . m,

where G r ( t ) i s a m o n o s p l i n e from t h e s e t ~ ~ d e f i n e d by f o r m u l a (3) by t h e nodes x k and t h e c o e f f i c i e n t s Pk"

THEOREM 5. The v e c t o r s ~ = {Xk} and ~ = {Pk} d e t e r m i n e t h e b e s t q u a d r a t u r e f o r m u l a of t he form (17) f o r t h e c l a s s W r L p , r i s a r b i t r a r y , l < p ( o o , i f and o n l y i f t h e v e c t o r s ~ and

d e f i n e a monosp l ine GrE d/~ and t h e numbers Xk, Pk a r e t h e s o l u t i o n s of t h e sys t em of equa- t i o n s (28) such x I (or Pl) is the smallest among the first coordinates of solutions. More- over,

] rq~~ ] + = 1, (29) 8 ~ ( w % ) = - i T . " ~ + ~q, 7 q

where ~o is the solution of the system (28), corresponding to the selected solution {x~}, {Pk}- The quantity (29) has order O(m-r).

THEOREM 6. The nodes x k and the coefficients Pk of the quadrature formula of the form (17), best for the class WrLp, r = i, 2, .... 1 < p < =, satisfy the relations

o~<22,~<Z1~<~, o ~ < s k = 2 , 3 . . . . . m; [m/21 1 Y , A (~ - 2 ~ ~ = 2~ + 2 ' ~ = I , 2 . . . . . 1(, - 0 / 2 ] ;

(--1)v+r+m[ "~'v - 2 P~(1--2Xk)~--II~O' " - - 1 , 2 . . . . . it/2].

k=l

For p > I the inequalities are always strict.

Already from these theorems it follows that for r > 3 it is difficult to express ef- fectively the nodes and the coefficients of the optimal quadrature formula of form (17) for the classes WrLp. Therefore, in the nonperiodic case the optimization problem of quadrature formulas loses its actuality if we look at this problem from the point of view of computa- tional practice, and the interesting problems of the theory of spline-functions enter in the first plan. In connection with this in the nonperiodic case there present interest the . quadrature formulas for which on the nodes or on the coefficients one imposes certain conditions (or are simply fixed), facilitating their investigation and their use. Examples of such quadrature formulas are the Markov quadrature formulas, mentioned in Sec. 2, the Sard quadrature formulas (see [3], Sec. Ii). Here belong also the quadrature formulas, considered in [43] by Velikin:

f (x) dx = ,,., '-" ~,J"> (x.) + ~ (/; X), (30)

where t h e v e c t o r of t h e nodes i s X = {0 = x o < x l ~ . . . < x r n - - - - - 1 } , w h i l e t h e c o e f f i c i e n t s Pk~ = ~ks are determined by the equalities

) I ~ l ~ I. l-{-1 Poz = b,h') + ' -Pro,, ----- ( - - , , . . . . . . . t = O, 1 . . . . . ,o,

20

ph~=b, l h ~ - - ( - - l ) Z h ~ ' l , k = 1,2 . . . . . m - - l , l = 0 , 1 . . . . . 0,

- -xk- i and bz-- p § 1 ~-I ( 9 + s ) I ( lq-s) l where hh Xh 1[ ~ " s l ( p + l + s @ 2 ) l '

S~O

The following theorem has been proved in [43].

THEOREM 7. Among the quadrature formulas of the form (30), where p = [(r - 1)/2], [(r - 1)/2] + 1 ..... r - i, the optimal with respect to the nodes on the classes WrLp, r = I, 2 ..... l~p~eo, is the formula with equidistant nodes x k = k/m. Moreover, the error of the optimal formula is determined by the equality

i~f s u p ~ ( / ; X ) = 1 1 [1 d 2~ 11 m'(29+2)' dt 2~176 X [ ~ Lq

1 1 ~ I . P q

In [43] one has o b t a i n e d a s e r i e s of o t h e r r e s u l t s r e g a r d i n g t h e e s t i m a t i o n and t h e o p t i m i z a t i o n o f t h e r e m a i n d e r ~ ( f ; x) o f t h e q u a d r a t u r e f o r m u l a (30) on t h e c l a s s e s o f 1- p e r i o d i c f u n c t i o n s ~ i r - 1 and WirL.

At the conclusion of this section we consider the problem of an optimal method of integral restoration.

Let O~x1~x=~..o~xm<l be arbitrary points of the segment [0, i], which, as it follows from the inequalities, may coincide. We assume that the information on the function fEC r-~

F lvv where 0~vt.~<r--l. Among all quadrature formulas of the form is given in the form -~xO,

I f(x)dx= p~f'~}(x~)+~) (31)

one h a s t o f i n d t h e b e s t f o r a c e r t a i n c l a s s H, c o n t a i n e d in C r - 1 The f o r m u l a t i o n o f t h i s p r o b l e m i s due t o K o l m o g o r o v , w h i l e f o r m u l a (31) i s c a l l e d t h e Kolmogorov q u a d r a t u r e f o r m u l a .

Thus , o n l y t h e number m o f u n i t s o f i n f o r m a t i o n i s f i x e d and a l l t h e r e m a i n i n g q u a n t i - t i e s ( o c c u r r i n g in t h e q u a d r a t u r e sum) can be d i s p o s e d o f f r e e l y . L igun [44] h a s p r o v e d t h e following theorem.

THEOREM 8. Among all the quadrature formulas of the form (31) for a fixed m, tlhe best for the classes WrL, r = i, 2,..., is the rectangle formula (to within a rigid shift of the nodes).

Zhensykbaev [45] has generalized Ligun's theorem to the classes W~Lp, l<p~oo, and has solved in a sufficiently general form the problem of the optimal choice of information for

1

the restoration of the integral S w(t) f(t)dl for the classes WrLp and WrLp, where the function 0

w(t) is almost everywhere positive and integrable on the segment [0, I].

6. Best Quadrature Formulas for Classes of Functions Defined by Rearrangement Invariant Sets. In the periodic case, deep and the most general results regarding the optimization of quadrature formulas have been obtained by V. F. Babenko. Of great interest are also the methods of investigation, developed and used by him.

Let Lp be the space of iv-periodic functions with the usual norm and assume that the numbers ~, B are positive. For any f6 Lp we set

where f+ ([) = max {~- f (~, O}.

By We m=.~ we denote the class of functions, the (r - l)-th derivative of which is absolutely continuous, whileIlf(~IIme.~l, and by q~,~ (~,~;t), m = i, 2 ..... r = 0, 1 ...... we denote a 2~/m-periodic function from the class Wp~lj~.I/~ with mean value on the period equal to zero and for which

where ~U = ~ ( 2 ~ / n - ~ ) .

21

Considering the quadrature formula

~" i ~ ~ f") (~) ~ ~ (f; x , P) (~2) [ i (t) ~ = ,

on a n o n s y m m e t r i c c l a s s o f f u n c t i o n s H c c r -~ ( f o r e x a m p l e , f o r ~ ~ ~ t h e c l a s s ~"~:~., i s n o t c e n t r a l l y s y m m e t r i c ) , i t i s a d v i s a b l e t o i n t r o d u c e t h e f o l l o w i n g c h a r a c t e r i s t i c s o f q u a d r a t u r e formulas:

R +(H: X, P i = s u p , ~,,:; X,P) , ; E L ~

R- (H; X, P) = inf ,'i' (f; X, P). ,EH

The p r o b l e m o f t h e b e s t q u a d r a t u r e f o r m u l a o f t h e fo rm (32) f o r a n o n s ~ m e t r i c c l a s s H c o n s i s t s i n t h e d e t e r m i n a t i o n o f t h e q u a n t i t i e s

~+ fH) inf R + (H; H, P),

gT.~/H~ = sup R - (H; X, 9). X , r,

and of the vectors (X +, P+), (X-, P-), realizing the corresponding infimum and supremum.

Babenko [46, 47] has proved the following statement.

THEOREM 9. Among all quadrature formulas (32) for p = 0, i, the best for the classes ~:~.~ for all ~, ~ E(0, ~ and r = I, 2 .... is the rectangle formula. Moreover,

c( 18~,o -r ~ 1 (w~;~.~)t = { ~ , , ( ~ ; ~ . ~ ) l = ~ ~ . r ~ , I~ ;" ) ) ' /71

where E • i s t h e b e s t l o w e r (E +) and u p p e r ( E - ) a p p r o x i m a t i o n by a c o n s t a n t in L1 o f t h e function f(t).

Extending the set of the classes of differentiable functions, V. F. Babenko has introduced classes of differentiable functions, defined by H-invariant sets, and has considered for these classs the problem of the optimization of quadrature formulas.

Let r ~; t) ~E LI, f(1)~0) be the decreasing rearrangement of the function f(t) on the

period. For any function f ELI we set

H ~; t) = r (f+; l) - - �9 (f_; 2~-- t ) , 0 ~< ~ ~ 2 ~ .

A set F c Ll is said to be rearrangement invariant (~-invariant) if from ~E F and ~(g; t) = H(f; t) there follows g~ F.

Rearrangement invariant is the unit ball in any symmetric space of 2~-periodic functions, imbedded in LI, in particular, in the L~, l~p~, Orlicz, Lorenz, Marcinkiewicz spaces. H-Invariant are the sets

2 ~

F (m) = {[ E L, ; ~ ~( lf (t) l )dt ~ 1}, 0

where ~ (t) 5s an a r b i t r a r y , n o n n e g a t i v e , n o n d e c r e a s i n g f u n c t i o n , d e f i n e d on [0 , ~ ) , e t c .

By WrF we denote the set of functions whose (r - l)-th derivative is absolutely continuous, while [(r~ E F.

We present two results from V. F. Babenko's investigations, related to the optimization of quadrature formulas on the classes WrF.

THEOREM I0 [47]. Let F be an arbitrary H-invariant set. Among the quadrature formulas (32) for p = 0, I, the best for the class WrF, r = i, 2 .... is the rectangle formula.

THEOREM ii [48]. Lt F be an arbitrary H-invariant set. Among all possible quadrature formulas of the form (31) for a fixed m, the best for the class wrF, r = i, 2 .... is the rectangle formula.

In [46-48] one has obtained the error of the rectangle formula on the classes WrF.

22

The results of the investigations on the optimization of quadrature formulas for periodic classes of functions have created the impression that the rectangle formula will be optimal for any class of periodic functions. However, Oskolkov [49] has shown that this is not so. Therefore, V. F. Babenko's investigations are valuable also due to the fact that in them one has extended the boundaries of the classes of functions for which the quadrature formulas with equal weights and equidistant nodes are optimal. We give one more class of functions for which this holds.

We denote by K*F the class of functions f(t), representable in the form

f if) = a v + (K,,p) (t), ( 3 3 )

where (K*q~)(t) is the convolution of the kernel KeL, and of the function c p E F ~ s orthogonal to H, where U = 1 if K(t) is orthogonal to a constant and U = 0 otherwise, while a is a constant.

A continuous kernel K(t) is called a CVD-kernel, written K6 C VD, if every function of the form (33), where q~ is continuous, has at most as many sign variations as ~ (t).

By J~ we denote the set of the pairs of vectors (X, P), defining the quadrature formula (18), such that Pl + P2 +''' + Pm = 2vo.

THEOREM 12 [50]. Assume that /<ECgDand that F is a lq-invariant set. If K(t) is orthogonal to i, then among the quadrature formulas of the form (18) the best for the class K*F

2~

is the rectangle formula. If [ /((1)dt~=0, then for any o, among the formulas (18), for which 0

(X,P) 6~ the best is

Z~ m

0 ~ 1

In [50] one has computed the error of the optimal quadrature formula on the class K*F. Theorem 12 generalizes in an essential manner the results of Grankina [51], Chakhkiev [52], Tkhi Tkh'eu Khoa [53]. At the proof of Theorems 10-12 one has developed new methods, with the aid of which one has established extremal properties of monosplines, that are deeper than those obtained by A. A. Ligun and A. A. Zhensykbaev. In [54], Babenko and Grankina have obtained optimal quadrature formulas for the classes K,F, where F is a unit ball in L I, while on the kernel K one imposes weaker conditions than in Theorem 12.

7. Quadrature Formulas for Functions that are Integrable with a Weight Function. In [55-57] one has considered quadrature formulas of the form (13) for the weight function q(x) = (I - x=)~, ~ > -I, whose vector of the nodes coincides with the set of the zeros F~ of the Jacobi polynomials of degree m, corresponding to this weight. For 8 E [ - -1 /2 , 1/2J one has obtained quadrature formulas of the form (13), asymptotically optimal with respect to the coefficients for fixed vectors of nodes X m = F~, and one has established the asymptotic behavior of these formulas one the class H m. In particular, it has been proved that the Gauss quadrature formulas, corresponding to the weight functions (i - x2) -I/2 and (i - X2) I/2 , are asymptotically optimal with respect to the coefficients for the classes H a.

We recall that the sequence of quadrature formulas with vectors of nodes X m and vectors of coefficients pm is said to be asymptotically optimal with respect to the coefficients for fixed vector of nodes for the class H if

where

Rq.m(H; X ~, P'~) = Rq.= (H; X') + o(R,.,~ (H: v ..... ,~ )J,

Rq,,~ (11; X m, # % = sup Ro (f; X ~', P'~), f E H

R q , , . (H; X% = inf Ro.m (H; X m, P'~). pm

Correspondingly, a sequence of quadrature formulas (13), determined by the vectors of nodes X m and the vectors of coefficients pm, is said to be asymptotically optimal if

Rq,m(H;X ~ ,P%=@~,m(H)+o($~ .... (H)),~gq.m(H)= inf Ro.~(H,X ~ , V ) . prn xrn

23

In the sequel we shall assume that i) q(x) > 0 almost everywhere; 2) q(x) is continuous on the interval (-i; i); 3) q(x) is monotone in some neighborhoods of the points -1 and 1 if q(x) is unbounded there.

We also note that q(x) is always assumed to be integrable. An example of a weight function, satisfying these conditions, is the classical weight q (x) = (l--x)~(l + x) ~, ~, ~>--[ We show that for unbounded weight functions, satisfying conditions 1-3, equality (16) holds.

Let X m be a sequence of vectors, defining a sequence of asymptotically optimal or optimal quadrature formulas of the form (13) for the class H ~. From condition 1 for the nodes x~ there follows the limiting relation

max hx~?-~O, (34 I ~ k ~ m

where AX~ x~+~ x ~ ~, x~ 1, ~ 1, k 0 ,1 , ,m. ~ --____ Zm+ I ~ ~ ...

This assertion, as well as its proof, is similar to V. A. Steklov's theorem [58, p. 627 of the Russian edition].

Making use of (15) and omitting the upper index in x~, we represent the error of the quadrature formula (13), optimal with respect to the coefficients and corresponding to the vector X TM on the class H a, in the form

i t x ~ q (X) dX dr_ m--lxkt'+~, d x i R,.~ (tP, x m) = ..~'- !" q (x) l x - x~f ~ ax = (x~ - _ ~ (x) ~ (x) + (x - xm) ~ q (x) dx, Ie~I ,4 k --1 k=l xf, Xm

w h e r e ~k (x) = ra in {x - - xk) =, (x~+, - - x) =, x 6 (x~, xk+0.

The first and the last terms have order o(i/m ~) and the sum of the remaining ones, making use of the continuity of the weight function, can be transformed with the aid of the mean value theorem in the following manner:

E ~ (x) q (x) dx = q (~) Ax~ +~, ~-k E (xh, x~+~). ~'~1 x k ,"~1

Thus, we have arrived at the known problem [59] of the minimization with respect to x k of the SLIm

ra--1

~-~ q (~k) (Xk+i - - X0 =+t. ( 35 ) d.d

m--!

~ (~) Ax? +' = ( m - - I) z_.

The upward convexity of the function t I/(~+I) for ~ > 0 allows us to estimate the sum (35) from below:

0 (~) Ax~ +~ k=l

m--I I

m - - 1 ( m - - l ) ~ ~=~

In the parentheses we have the Riemann sum for the function ql/(~+1)(t), which for m ~ tends to the integral of this function, due to the conditions imposed on q(t) and to the limiting relation (34). Thus,

l

R,.m (H~; 8m (36) X = ) > I ( ~ + 1)(2m) ~ _~ t

w h e r e ~m = o ( 1 ) . I n o r d e r t o o b t a i n a n u p p e r e s t i m a t e f o r ~,.m(H~), o n e h a s t o c o n s i d e r a sequence of vectors of nodes X m = {x~} such that

24

x, Xk+l !

.I q (x) (x 1 - - x)=dx = S q (x) q~h (x) dx = S q (x) (x - - .,:~,)a dx .--- Cm, k = 1, 2, . . . . m - - 1. ~ I ~: k Ztn

In this case the sum (35) is equal to

l m - l l

I t i s e a s y t o show t h a t Cm..~o and t h a t (34) h o l d s f o r x~o C o n s e q u e n t l y ,

1 1

( e + 1) (2m) ~ ~ ( 0 d + - - e l

where 8 m = o ( 1 ) . From t h e i n e q u a l i t i e s ( 3 6 ) , (37) t h e r e f o l l o w s

t 1

(7) 1 (t) a t + o , g~,m (It ~) = (~. + 1) (2m) ~ --1

where t h e o r d e r o f t h e s e c o n d t e r m i s d e t e r m i n e d by t h e f u n c t i o n q ( t ) and t h e number ~.

(37)

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25

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Documents

Investigation of the optimization of quadrature formulas by Dnepropetrovsk mathematicians