Stieltjes-type functions and elementary quadrature formulas (I)

              Rev.Acad . Canar.Cienc., III (Num . 1) , 65-74 (1991)
STIELTJES-TYPE FUNCTIONS AND ELEMENTARY QUADRATURE FORMULAS (I)
Juan C. Santos Le6n and Pablo Gonzalez-Vera.
Dpto. Analisis Matematico. Universidad de la Laguna.
38271 La Laguna. Tenerife. Spain
ABSTRACT: In this paper we study certain rational approximations to functions
of Stleltjes-type;l. e. functions that can be given by (1.1). For it, we shall
make use of some well known quadrature formulas, such as, the trapezoidal
rule, the mid-point rule an so on. Numerical examples, where comparison with
others methods is made, are also given.
Keywords: Stieltjes functions, quadrature formulas, Pade approximant, uniform
convergence.
A.M.S. Classification: 40C, 41, 65D.
1. Introduction
Historically, in the last one hundred years, many authors have been interested
m· s t u d ym· g f unc t"1 0ns o f th e f orm, g(z)-- Jb dza(- xx ) for some b ound e d , non
a
decreasing function a(x), I do: I= J~ da(x) >O and where a, b are in general,
extended real numbers. We say that g(z)eM(a, b). Take into account that a
simple change of variable allows us to write g(z) in terms of the function
f(z)=Jb da( x) (1.1)
a 1-zx '
and it will be said that f(z)eS(a,b).
Remark 1. West European authors often use the clas S(a, b) of the Stieltjes
functions while Russian authors use the class M(a, b), the functions of which
they call Markov-functions.
In what follows, we shall be concerned with the class S(a, b), whose name is
due to the mathematician F. J. Stieltjes who in his memoirs from 1894,
"Recherches sur les fractions continues" [l] studied certain continued
fractions of the form
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az + a+ az + a+
(1.2)
1 2 3 4
with a?O; i=l,2,3 .. .. , proving that (1. 2) converges to a function
f(z)=J00 da( x) where a(x) is a bounded non-decreasing function. Since then,
o z+x
many contributions in this context have been given in connection mainly with
"Moment problems",and where topics such as, Pade approximants, Quadratures
formulas, and orthogonal polynomials arise in a natural way. A good historical
review can be found in [2]. Among the most recent works concerned with the
estimations of (1.1) we can mention for example, the papers by M. Prevost (3)
and P. Gonzalez-Vera (4), where the generating function (l-xz f 1 (x variable;z
parameter) is replaced by an adequate interpolatory polynomial o Laurent
polynomial ( see also [S)). As it is well known, this is the basic idea of the
so called "Interpolatory quadrature formulas" in order to estimate a definite
integral. In this sense, the starting point of our work consists of making
use of certain elementary quadrature formulas, that allow us to provide
readily computable estimations of functions which can be written in terms of
those ones belonging to the class S(a, b) with a, b two finite real numbers.
2. Theoretical Result
Let us assume n a fixed natural number (n?:l) and consider the uniform mesh of
the finite interval [a,b), a=x
0
<x
1
< ... <x
0
=b; that is: x
1
=a+ih with h=(b-a)/n.
Let R
1
(x, t) be the polynomial of degree one at most, which interpolates the
generating function l-xt at the points of the mesh x
1
_
1
and x
1
. One has,
[
x-x .
R (x t)=-1- ___1- _l_
l ' h 1-x t
l
x-x
l
1-x t
l -1
so that, by writting f(t)=Jb da( x) = r (i
a 1-xt L J x
. l =l 1-1
Rn ( t):= f (l R (x,t)da(x) L Jx 1
1 =l 1-1
represents an estimation of f(t) .
66
i=l,2,3, . .. ,n.
da(x)
1-xt
then
(2.1)
(2.2)
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Proposition 1. R (t) is either a n/n+l rational function, that is, th1e
n
denominator is a polynomial of degree n+l at most an the numerator of degree n
at most, with poles J l/x. L ~ l I ( 1=0
if x *O i=O,l, .. ,n; or a n/n rational function
i
with poles ~ l/x ~n if x =O.
l l=O,l*k k
PROOF. Indeed, replacing (2.1 ) in (2.2), one gets:
B
l
1-x t
1-1
where A = ( 1 (x-x )da(x) and
I J x 1-1
1-1
l=l
B = ( 1 (x-x )da(x). Therefore,
I J x l 1-1
B
n-1
A -B A
R (t)~(- 1 +l I 1+1 n ) (2.3)
1-x t 1-x t
+
n h 1-x t
0 1 n
1=1
Thus, from (2.3) the proof follows immediately.
Remark 2 . Recall that this procedure to obtain R (t) is the well-known
n
"Repeated Trapezoidal Rule" to approximate J~ F(x)da(x) with a(x)=x.
On the other hand, the function f(x)eS(a, b), is holomorphic in the cut plane
OJ
D(a,b)=Jze!C:z~ [a- 1,b- 1
]L and the series f =\cl · with c =Jb xkda(x) is an
l r olk' ka
k=O
asymptotic expansion of f(t) at t=O (see [6]). In this sense, one has
Proposition 2. The rational function R (t) defined by (2.2), satisfies:
n
OJ
f(t)-Rn (t)=O(t2)= l sk l (n~l)
k=2
PROOF. We must prove that R (t)= c +c t+O(t2
) . Indeed,
n 0 1
n A
Rn(t)=+[L -1---x-\-
1 l=l
OJ n
I= 1 I= lj=O
=-h
1
L\ ( L\ A xJ -B xJ ) tJ. For j=O, we have: 11 I 1-1
j=O l= 1
l=lj=O
1 ~ (A -B )~ ~ I x
1
(x -x )da(x)= ~ I x
1
da(x)=c h L I l h L l 1-1 L 0
x x 1=1 l=l 1-1 l=l 1-1
otherwise, for j=l, it yields
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n n
[ x I x, (x-x )da(x) - x I x, (x-x )da(x)] 1 = l (x A-x B )=-
1
h -I l l 1-1 l h l 1-1 1-1 l
x x
l=l l=l 1-1 1-1
n
(x -x ) I 'i xda(x) = J: xda(x)=c
=+I 1 • I i-1
x
l=l 1-1
In order to assure the pointwise and uniform convergence of the sequence
~Rn (t) ~ ·to f(t), the following lemma will be required. The proof is a direct
consequence of Balzano' s Theorem.
Lemma 1. Let g be a continuous function on the finite interval [a, b] and a, (3
constant such that a{3>0. Then, there exist ~e[a,b] so that:
ag(a)+{3g(b)=(a+{3)g(~)
Theorem 1. The sequence ~Rn (t) ~ converges to f(t) for any telC-[a-1, b-1).
PROOF. For telC-[a-
1
,b-
1
] the generating function (l-xtf1 is a continuous
function on the interval [a, b]. On the other hand, the Riemann-Stieltjes
. Jb da(x)
integral a ~ exists.
By applynig Lemma 1 on each interval [xl-1,x
1 l (take into account that A?O
n A
and -B
1>0l one gets: Rn(t)=-hl \ (
1
L l-x
1
t
l=l
;~ 1e[xl-1,x/ From the definition of \ and B1 one can write:
A= ha(x )-I x
1
a(x)dx and B = ha(x ) - J x
1
a(x)dx
l l l 1-1
x x 1-1 1-1
Therefore, A -B =h[a(x )-a(x ) ]=hLla(x )=hLla
I l l 1-1
and Rn(t)= f
l=l
proof follows, as n tends to infinity.
f1a
l
1-~ t •
I
Now
Theorem 2. The sequence ~ R (t) ~ converges uniformly to f on any compact
n
n B c nl n
the
A Cnl
PROOF. I Rn (t) I :S+ l I I l=+l IA<n>_ Ben> I ;(2.4)
1-x t 1-x t I I I 1-~Cn\ I l 1-1
l= 1 l=l l
where ~(n) [ (n) (n)]
1 e x1-1'x1 · Now, a constant B=B(K) dependent on K exists such that
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11-xt I :?:B(K)>O. Therefore, from (2.4)
c
0
IRn(t)l::s B(K)' On the other side, for
any n, R (t) represents an analytic function in IC-[a -l, b-\ because of its
n
poles ix~ 1 ~ c[a-1,b-1
]. It follows that iRn(t)~ is a normal family in
IC-[a-1,b-1
] and hence by the Stieltjes-Vitali Theorem (see, e.g. [7, Theorem
15.3. l]), the proof follows.
Remark 3. The above results can be easily extended to other elementary
quadrature formulas. Thus, for the "repeated midp-oint rule" one has
f(t)=Jb da( x)= ne (1+1 da( x)
a 1-xt l Jx 1-xt
l=O I
n-1 x "" \ J 1 + i da ( x)
l x 1-y t
I =O i i
n-1 A
= l~:=Mn(t)
I =O YI
x+x fx 1 1+1 b-a 1+1 where y
1 2 ; xi=a+ih; h= n ; i=0,1,2, .. . ,n; and A
1
= da(x)
l
For the approximating function M (t), the following theorem holds
n
Theorem 3. The sequence ~Mn (t) ~ of rational functions with poles at y ~ 1
satisfies:
(i) M (t)=c +O(t).Furthermore M (t)=c +tc +O(t2 ),in the particular case a(x)=x
n 0 n 0 1
(ii) iM (t)~ converges uniformly to f(t) on any compact K of !C-[a-1,b-\
n
3. Numerical Applications.
In this section the above procedures are tested and a comparison with others
methods of similar characteristics, is made. For this purpose, we have chosen
the following which were considered by Prevost (see, [3]).
1) Let B (x, t) be the corresponding Bernstein polynomial of degree n on the
n
interval [a,b] for the function (l-xtf1 (as usual, x variable and t being a
parameter); then H (t)=Jb B (x,t)do:(x)<>< f(t).
n a n
H (t) is, certainly a rational, satisfying
n
(i) f(t)-H (t)=O(t2
)
n
(ii) The sequence ~H (t)~ converges to f(t) for any real t~[a-1 ,b-1 ].
n
2) Let P (x, t) be the interpolating polynomial of degree n to the function
n
(l-xtf1 at the equidistant nodes x
1
=a+ih; then, R (t)=Jbp (x, t)do:(x) is a
n a n
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rational function satisfying f(t)-R (t)=O(t0
+
1
).
n
As it is well known, R (t) is a (n/n+l) Pade-type approximant to the formal
00
series f = l c tJ
0 j =O j
n
with c =Jb xJda(x), such estimation can be seen as a
J a
Newton-Cotes formula (For more details concerning with Pade-type approximation
see Brezinski [8]) .
We first consider the test funct1. 0n - -t1- log(l-t)=J 1 dx
0
l-xt (a(x)=x;[a,b]=[O,l])
In this particular case we have also made use of an improvement of the
trapezoidal rule, namely, the so-called "corrected trapezoidal rule", given by
2
([9]) T~(t)=T0(t)- ~2 [g'(b)-g'(a)] (g(x)= 1-xt
For Tc(t), the following statement holds true.
n
Proposition 4. Let Tc(t) be the rational approximating function obtained from
n
the "Corrected Trapezoidal Rule", when a(x)=x, then f(t)-Tc(t)=O(t4
).
n
c h
2
[ d 1 d 1 ]
PROOF. T (t)=T (t)- -12 -d (-1--t-) I - -d ( -1 -t-) I = n n X -x x=b X -x x=a
oo n oo oo
2 2
=-1-
\ ( \ A xj -B xJ ) tj - _h_ \ .bJ-\J + _h_ \ ·aj-ltj
h L L I I I 1-1 12 L J 12 L J
J=O I= 1 j=2 J=2
thus from Proposition 2 one has, at least, f(t)-Tc(t)=O(t2)
n
n
For j=2, it must be shown that--/;-l A1x: -B1x:_, - ~~ 2b + ~~ 2a=c2=J:x2dx
i=l
Now, by replacing the value of A
1
and B
1
in the above expression, it results
because of the corrected trapezoidal rule is exact
for polynomials of degree 3. In a similar way, it can be easily
n
1 \ 3 3 h
2
2 h
2
h L 2 Jb 3 A
1
x
1
-B
1
x
1
_
1
- l2 3b + l2 3a =c
3
= x dx
a I= 1
The numerical results are now displayed in the following tables.
70
seen that
•
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t=-1
t=-2
n
t=-1
5
10
15
20
25
30
t=-2
5
10
15
20
25
30
t=-5
7
14
20
t=-8
7
14
20
H ( t) T ( t) M ( t)
n n n n H (t): Bernstein
1 0.75 0.75 0. 66
2 0 . 722 0.708 0.685
n n
T (t): Trapezoidal
n n
3 0.712 0.7 0. 689 M ( t): Mid-point
4 0 . 707 0.6970 0.6912 n
5 0 . 704 0.6956 0.6919
10 0.6988 0.693771 0. 6928
20 0.6960 0.693303 0.69306
30 0.6950 0.693216 0.693112
log 2 0 . 69314718 ..
1 0.66 0.66 0.5
2 0.61 0.583 0 . 533
3 0.5904 0.5650 0 . 5416
10 0 . 5613 0 . 55078 0 . 54857
30 0. 5532 0.549470 0. 549223
10~ 3
=0 . 549306 ..
Table
Pade-type:R ( t)
n
Corrected
Trapezoidal: Tc
n
. 69317 4603 . 6931349
. 6 931 4 7 2 0 2 7 8 . 693146403
. 69314 7 1805631 0 . 693147026
.6931471805599508 . 69314 7 13179
. 6931471805599453 . 693147160575
. 6931471805599454 . 693147170920
log 2= . 6931471805599453
. 5496296 . 5492078
.549307685 . 5492996
. 549306147194 . 549304854
. 54930614436334 . 5493057347
.5493061443341384 . 549305 97 62
.5493061443539317 . 54930606321
log(3)/2=.5493061443340549
. 3589619 . 35799
.3583576083 . 35832626
. 35835 20086 . 358345570
log(6)/5= . 358351893
.276580 . 2734300
. 27469998 .2745553
. 274655390 . 2746281
log(9)/8=.27465307216
SIMPSON
. 693150
. 693147374
. 693147 219
. 693 1 4 7 19274
. 693147185554
.693147182969
.54933029
. 54930775
. 5493064660
. 5493 0624660
. 549306186312
. 5493 06164602
. 35843781
. 358358212
. 3583 53463
.2749304
. 274676721
. 2746591913
Table 2
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n R ( t) SIMPSON T ( t) M ( t)
t=-12 n n n n
7 .217913 .210546 .214421 .2308 .2062
15 . 2138510 .2135177 .21379964 .217935 . 2 11 73
30 . 2167422 . 21372917 .21374985 .214833 . 213207
40 60.78 .213740376 . 213747117 .214361 .213439
log ( 13) I 12=. 213745779 ..
t=-20
7 . 16090 . 14284 . 153952 .17678 . 1 4253
15 . 15270796 . 1513943 .15241037 .158784 . 1 49223
30 . 18126 .1521563 . 152242844 . 154004 .151362
40 7.6150 . 15220259 . 152231853 .153241 .151726
1 o g ( 21 ) I 2 0 = . 15 2 2261 ..
t=-30
7 . 12 782 .094618 .1176056 .145585 . 1036
15 . 115648 .112367 .1148940 . 123466 . 1106
30 .233 .114262 .1145133 . 117037 . 1132
40 12984 . 3 .114394 .11448333 . 115955 . 1 137
log(31)/30=.11446624 ..
t=-150
7 .0662 - . 16 . 045 72 . 0877 . 024689
15 . 0411 -.001299 .036997 .054253 . 028369
30 1.6294 .0274236 .0343391 . 041311 . 030852
40 6891. 03 .0307088 .0339059 .038521 . 031598
1og(151) I 150= . 0334485 ..
Table 3
In table 1, our estimations T (t) and M (t) are compared with that one
n n
provided from the use of the Bernstein's polynomials. The three rational
approximations have the same order.
In tables 2 and 3 comparison with Pade-type approximations R (t) with
n
generating polynomial v(x)=(x-x ). .. (x-x )
0 n
( ~x ~n
I 0
being equidistant on the
interval [O,l]) is made. This is equivalent to make use of Newton-Cotes
formulas; and, as it is well Known, such formulas may diverge, even when an
analytic integrand function is considered. The following result (see [10])
may throw light on the numerical results appearing in tables 2 and 3.
Theorem 4. Let the Ellipse £? be whose center is at (a+b)/2, whose major axis
lies on the x axis, and whose semimajor and semiminor axes have lenght
8 5 3 (b-a) and s(b-a), respectively. If f(z) is regular in the clousure of
f5 then the Newton-Cotes quadrature formulas over [a,b) will converge to
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f:f(x)dx, as n-7 co. The convergence is of geometric rapidity.
From Theorem 4, the convergence of the sequence ~Rn (t) r is assured for
tE(-8,0.8); futhermore such sequence converges geometrycally. On the other
hand, for te(-8,0.8), the numerical results show the divergence of ~Rn(tlr.
however our estimations still give rather good results.
Nextly, the test function to be considered, is fw du
Arctan w- --- , which can
- o l+u2
be written in the form Arctan w= __ w __ G(-
1-
-1)
2(1-w2
) w2
where,
n R ( t) H ( t) n T ( t) M (t)
t=-2
n n
t=-2
n n
5 .67647 .6999 5 . 6834 .6471
15 .67551087485 .6837 10 . 67773 . 6643
30 . 675510858856498 . 67963 20 .67610 .6713
50 .675510855569 .67798 25 . 6758 97 . 6724
70 .67551031712 .388 50 .6756112 .6744
85 .67691 . 276 75 .6755561 . 6749
E.V.= .6755108588560400 ..
5 .5228
15 .5144220 [ .55374211
25 .51441282259 .5262
40 .5144265 7 .5218
55 9.948 .51979
65 .387E+l0 -41 . 08
t=-5
5 . 5388 . 458
10 . 5220 . 4903
25 .51585 .5073
50 . 514798 .5117
75 . 514588 .5129
t=-5
E.V.= . 514412800 ...
5 .4538
[
.517
15 .43530 .4637
30 .43521157 .4494
40 .4389 .4458
50 -2662.2 .4437
70 -1 . 9E+l0 .2993
t=-8 5 . 4747 .362
15 .4420 .4144
20 . 43930 . 4209
30 . 43716 . 4269
50 .435961 . 4311
75 .435556 . 4329
t=-8
t=-12
E.V.= . 4352098 ...
t=-12 5 . 428 .285
5 .404 .478 15 . 3833 .344
15 .37277 .410 20 . 37913 . 3527
20 . 37239637 .4007 35 .37485 . 3628
25 .3723347 .3950 45 . 37392 .3655
35 .3672 .3885 55 . 37342 .3671
45 73.60 .3849 75 . 37293 .3689
75 -2.85E+l7 -52.56
E.V.= . 37232205 ...
Tab 1 e 4 Table 5
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2
G(z)=zf
I
(x-1)-112 dx
z+x
after making the succesion of transformations, u=wv,
introducing the function H(z)= f 2 (x-1 )-112 dx
1-xz
one can write G(z)=H(-1/z).
I
2
Therefore, Arctan w= w H(--w-) and where H(z) has the form of the
2(1-w2
) w2 -l
integrals (1.1). The numerical results for this case are given in Tables 4
and 5.
References
[l] T. J. Stieltjes, Rescherches sur les fractions continues, Ann. Fae. Sci,
Toulouse 8 (1984), J , 1-122; 9 (1898), A, 1-47; Oeuvres 2 402-566.
[2] J. Gelfgren, Rational interpolation to function of Stieltjes-type, Univer­sity
of Umea: Departament of Mathematics. Technical Report, N. 6 (1978)
[3] M.Prevost: Sommation de certain series formelles par approximation de la
fonction generatriz. Tesis. Universite de Lille (1983).
[4] P. Gonzalez-Vera and L. Casasus, Two-point Pade-type Approximants for
Stieltjes functions.Lecture Notes in Mathematics,N. 1171;(1984) pp 408-418
[S] W.B. Jones, 0. Njastad and W.J. Thran, Two-point Pade expansions for a
family of analytic functions, Jour. of Comp. and App. Math. 9 (1983)
105-123
[6) W.B. Jones, W.J. Thran and H. Waadeland, A strong Stieltjes moment
problem, Trans. Amer. Math. Soc. 206 (1980) 503-528.
[7) E.Hille, Analytic function theory, Vo! II. Ginn, New York, (1962).
[8) C. Brezinski, Pade-type approximation and general orthogonal polynomials,
Internat. Ser. Numer. Math. SO (Birkauser, Basel) (1980).
[9) K.E. Atkinson, An Introduccion to Numerical Analysis, Wiley Editions, Inc.
Singapore, 1988.
(10) P.J. Davis, On a problem in the theory of mechanical quadratures, Pacific
J. Math. 5 (1955) 669-674.
Re cib ido : 1 4 de Abri l de 1991
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