4. Proof of Theorems 1 and 2. Necessity. The Esseen and Janson theorem implies that (4) yields the upper bound in (5) and conditions (2) and (3). We shall now determine the lower bound in (5). To do this, we utilize Lemma i and show that (8) is fulfilled.
Set Yk = X2k - X2k-l, k = i, 2, ... We shall now verify that for this sequence (4) is fulfilled. The upper bound follows from the corresponding bound on {Xk} and the inequa- lity MIU + VJP ~ m a x { 2 P -l, i} (MJUJP + MIVIP) [2, p. 168].
If 0 < p < i, the lower bound follows from condition (6). For p e i, this bound fol- lows from (2) and the following well-known inequality [2, p. 277]. Let the r.v. U and V be independent, possess a finite absolute moment of the order p e i and MU = O. Then MIU + VlP e MIVIP.
The r.v. possesses the c.f. If(t)l 2. By Lemma 4 the bounds (8) are fulfilled for If(t)l 2 We have
I - - 1 / ( 0 1 = = (1 - - ; f ( O t ) O § tf(t) l ) ~ < 2 0 - - Re f (t)).
This yields the required lower bound for 1 - Re f(t). The upper bound in (8) follows from the above-established upper bound in (5) and Lemma i. Theorems i and 2 are thus proved.
LITERATURE CITED
i. C.-G. Esseen and S. Janson, "On moment conditions for normed sums of independent vari- ables and Martingale difference," Stochast. Process. Applic., 19, No. i, 173-182 (1985).
2. M. LoSve, Probability Theory, Van Nostrand, New York (1985).
3. N . N . Vakhaniya, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions in Banach Spaces [in Russian], Nauka, Moscow (1985).
4. V . V . Petrov, Sum of Independent Random Variables, Academic Press (1970).
ONE TYPE OF GENERALIZED MOMENT REPRESENTATIONS
A. P. Golub UDC 517.53
In 1981, V. K. Dzyadyk [i] first investigated the problem of generalized moment repre- sentations of numerical sequences, a problem which has found useful applications in the study of rational approximations and integral representations of functions.
Definition i. A generalized moment representation (GMR) of a sequence of complex num- bers {Sk}k=0 ~ is defined to be the set of equations
s~+i = 5 a, (t) bj (t) d~ (t), i, ] = O, oo, ( 1 )
X
in which d~(t) is a measure on a set X (X is most often a segment of the real axis), and a{ai(t)}i=0 ~ and {bj(t)}j=0 ~ are sequences of measurable functions on X for which all inte- grals in (i) exist.
In the present article an analogous construction is investigated based on the concept of q-integral introduced by Jackson [2].
Definition 2. For some fixed generally speaking complex number q, lql < i, the q-inte- gral of the function ~(x) defined on [0, i] is defined by the following formula:
CD(x)=~(p(u)dqu.=x(1--q)
xq~(xq'Z)q n,
xC[O, 1],0 n = 0 (2)
Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. ii, pp. 1455-1460, November, 1989.
p r o v i d e d the series on the r i g h t - h a n d side of (2) converges.
R e m a r k i.
e x a m p l e [ 3 ] )
then
It is obvious that if we d e f i n e the q - d e r i v a t i v e by the formula (see, for
dq ( D ( x ) = q ) ( q x ) - - ( 1 ) ( x ) ( 3 )
dqx ( q - - 1) x '
(u),dqu =-- ~ (u).
dqx 0
Thus, we i n t r o d u c e q - i n t e g r a l GMRs into consideration.
D e f i n i t i o n 3. A q - i n t e g r a l g e n e r a l i z e d m o m e n t r e p r e s e n t a t i o n of a s e q u e n c e of c o m p l e x numbers {Sk}k=0 ~ is defined to be the set of e q u a t i o n s
I
st+i = ~ a, (t) bj (t) dqt, i, ] = O, oo,
0
(4)
in w h i c h { a i ( t ) } i = 0 ~ and {bj(t)}j=0 ~ are s e q u e n c e s of f u n c t i o n s such that all q - i n t e g r a l s in (4) exist.
It is known [4] that GMRs of type (i) under specific conditions can be r e p r e s e n t e d in the form
sh = ~ (A~ao) (t) b o (t) d~ (t), k = O, ~ ,
X
w h e r e A is a linear b o u n d e d o p e r a t o r on a B a n a c h space. A s i m i l a r i t y t r a n s f o r m a t i o n is pos- sible also for q - i n t e g r a l GMIRs. In fact, we i n t r o d u c e two i n f i n i t e - d i m e n s i o n a l spaces of f u n c t i o n s ~J~ and ~ d e f i n e d on [0, i] such that M ~ E ~ and M ~ E ~ the b i l i n e a r f o r m
!
< % @ : = ~ q~(u)~(u)dqu = (1 - - q) ~" cp(q~)~(qn)q ~
0 n = O
(s)
is defined. We c o n s i d e r a bounded linear o p e r a t o r A :-:DI-+9~ and a s s u m e that there exists a u n i q u e linear b o u n d e d operator A * : ~ - + ~ s u c h that
We shall say that o p e r a t o r A* is c o n j u g a t e to A w i t h respect to b i l i n e a r f o r m (5). N o w if we a s s u m e that a0(1)Eg~, b0(1)E~ and linear o p e r a t o r A has p r o p e r t y (Aai)(t) = a i+1(t), i = 0, =, w h e r e ai(t), i = 0, ~ are the functions a p p e a r i n g in Eqs. (4), t h e n it is obvious as well that we w i l l h a v e
(A'b j)(t) = bi+~ (t), ] = O, ~ , a n d , c o n s e q u e n t l y , we o b t a i n e x p r e s s i o n s
1
sh = I (A~ao) (t) b o (t) dqt, k = O, co, (6)
e q u i v a l e n t to (4).
It is p a r t i c u l a r l y simple to carry D z y a d y k ' s t h e o r e m [5] on c o n s t r u c t i o n of d i a g o n a l Pade a p p r o x i m a n t s over to the case of q - i n t e g r a l GMRs. We shall state a m o r e g e n e r a l asser- tion h e r e that is true for a r b i t r a r y b i l i n e a r forms.
T H E O R E M i. Let an a n a l y t i c f u n c t i o n f(z) be r e p r e s e n t a b l e in a n e i g h b o r h o o d of z = 0 by the p o w e r series
f (z) = f (0) +
,V
s : ~+~, ( 7 )k=O
and for sequence {Sk}k=0 ~ let
s~+: = <a~, hi), i, / = 0, ~ ,
where "i e ~ ' i = O, 0% bj e 9, , j = O, o% ~ and 9~ are i n f i n i t e - d i m e n s i o n a l linear spaces, and <', "> is a bilinear form defined on the Cartesian product of these spaces. Further,
A ~ B
let there exist n o n d e g e n e r a t e biorthogonal sequences { M}M=0 , { N}N=0 :
M N
c i a i, CM --r ~ , M = 0, c o ; B N = _c,V c (N)I~i
~J,
i = 0 i~O
7V ~--- 0, 0 0 7
for which
<A~I, BN) = 6~hN, M, N = 0, oo.
Then the diagonal Pade polynomials [N/N]f(z), N = 0, = of function f(z) can be repre- sented in the form
A r
V clA'~z'~-:T: (:; z)
[N/NI/(z) -- :=o ~v ,
Z c}N)zN--]
/ = 0
where Tj(f; z) are partial sums of series (7) of order j.
Now we construct q - i n t e g r a l GMRs for certain functions.
Example. Consider the following function, sometimes called the exponent q-analog [6]
o0 oo
'+' 2 '%
k=O k=O
1 - - qk k
where kq= ~ , k q ! = ~ i v, 0qI:=l.
i=I
In order to construct the q-integral GMR for {Sk}k=0 ~ we use the definitions and pro- perties of q-gamma and q-beta functions (see, for example, [7]). The q-gamma function is defined by formula
where (a; q)~ = f i (1 -- aqn).
tZ~0
rq ( x ) - (q; q)---~= (l - - q / - 5 (qX; q)~
It is obvious that
rq(n-{-1)---- (1 --q) (I(l_q)n-q=)'''(1-q n) .=nq!.
The q-beta function is defined by means of q-integral
i
~ (qn+1; q)ooBq (x, y) - - - - t x-I (tq; q)~o dqt --- (1 - - q) qnX (qn+y; q)oo e
(tq~'; q)oo n = o (8)
and can be expressed by means of the q-gamma function by the formula
rq (x) Fq (y) B q ( x , y ) = F q ( x + y ) ' F r o m (8) and (9) we get
l
1 f t x-1 (lq; q)oo dqt.
l'q [x-i-y) -- Fq (x) Fq (y) (tqV; q)o~
0
Setting x = i + i, i = 0, ~, y = j + i, j = 0, = we a r r i v e at the q - i n t e g r a l GMR
1
s~'+i -- (f "v- i -+- l)q! - -
iq!
0
i--I
I--] (1 --
tq n+l)
r ~ O
iql dqt.
Thus, the u p p e r part of the Pade table for Eq(z) can be c o n s t r u c t e d in terms of poly- nomials Qn(t), n = 0, ~ that are q - o r t h o g o n a l on [0, i], i.e., that s a t i s f y the e q u a t i o n s
| oo
( Qn (t) Qm (0 dot = (I -- q) ~ Qn (q~) Qm (ql) qi____ 0
r'=0
for m = n. Such p o l y n o m i a l s have been r a t h e r w e l l studied up to the p r e s e n t time (see, for example, [3; 8, p. 92]).
G e n e r a l i z i n g the arguments given above, we get the f o l l o w i n g result.
T H E O R E M 2. For a s e q u e n c e {Sk}k=0 ~ of the c o e f f i c i e n t s of the power e x p a n s i o n of func- tion
oo o o
__r v
f (z) = s~zk = __, r,.~ (~le + v) '
k = O k = O
~,~>0, (i0)
a q - i n t e g r a l GMR of the form
l
sL+i - - rq (,ui + ~ / + ~) =
t ~ ' + v ' - '
(tq; q)~
Fq (~, -+- vl) F~ ( l a / + v2) (tq~J+~'; q)oo
d$,
(ii)occurs w h e r e vi, ~2 > O, v z + ~2 = v.
We n o t e that GMR (ii) admits an o p e r a t o r f o r m u l a t i o n of type (6) w i t h o p e r a t o r
and initial functions
1
x ~ ~ (tq; q).._______~
(Q~q~) (x) - - F,~(~O
(t@~; q)~ ,p(xt) d~t
t v'-l (lq: q)~
a o ( l ) = F q ( v 0 , b o ( / ) =
F~(v~)(tq~;q)~
( 1 2 )M o r e c o m p l i c a t e d q - i n t e g r a l GMRs can be c o n s t r u c t e d if instead of o p e r a t o r Q~ we con- sider operator
(Q~) (x) = x ~ (x) + ~ (Q~) (x)
w i t h t h e s a m e i n i t i a l f u n c t i o n s ( 1 2 ) . I n t h e p a r t i c u l a r c a s e u = 1 , v 2 = 1 , v l = v we g e t t h e f o l l o w i n g r e s u l t .
THEOREM 3 . F o r a s e q u e n c e { S k } k = 0 ~ o f t h e c o e f f i c i e n t s o f t h e p o w e r e x p a n s i o n o f f u n c t i o n
#--I
nlq- ]
- - - F ~ ( v + k + l)
I , ' = 0 ~ = 0
Z ~ ,
we get a q-integral generalized moment representation of form v > O ,
(+i--1
l--q + e
m=O ---
f
a~ (l) bj (t) d,,t,s~+i= p.(~ + i + i + 1) i , ] = 0 , ~ ,
where polynomials ai(t), i = O, ~, bj(t), j = O, ~ can be expressed by formulas
f--;
a i ( t ) = p . ~ ( v + i ) 1 - - q
I
bj(O=l ---F.' ij ( i -
at
(13)
(14)
(15)
in which
Proof.
eo
(z) = ~-~ (• - - q/~) (• - - q~.-T) .., (• _ 1) ( 1 - - qk)(l __qk-~)... (1 - - q )
le=O
Consider the operator
(QoqD) (x) = xCp (x) + ~ (Qq~) (x).
z , •
(16)
(17)
Construct its conjugate with respect to bilinear form (5)
I
S
(O.~) (x) = x ~ (X) + ~ ~ (u) d~u.
qx
(18)
It is rather simple to establish the fact that successive application of operator (17) to initial function a0(t) = t/F (v) reduces to formulas (14); moreover, the corresponding generalized moments will equalqthe coefficients of the power expansion (13). In order to obtain formula (15), first we construct the resolvent of operator (17). For this it is neces- sary to solve the q-integral equation
(x) - - zx~ ( x ) - - z~ i ~ (u) dqu = ~ (x),
0
which by an application of operator (3) reduces to the following q-differential equation:
(1 - - zqx) ~ dq ~ (x) - - z (1 -;- ~) ~ (x)
(o) = ~p (o).
= ~ ~ (x),
(19)
After applying the power series method, we shall establish that function ~(x) = ~(zx) defined by formula (16) satisfies the homogeneous equation. For solution of the inhomogeneous equa- tion we apply the method of variation of the constants
( x ) = C (x) ~ (x).
S u b s t i t u t i n g e x p r e s s i o n (20) i n t o Eq. ( 1 9 ) , we g e t
(20)
d,?
do C (x) = d,F ~p (x)
d j (1 - - zqx) X (qx) '
and, therefore,
d o
C (x) = C o -F ,j (1--zqu))~(qu)
i
d,;u r (u) dqu.0
As a result we get the formula for the resolvent (x) q~ (zx)
9 q: ( . r ) =
(R~Qocp)
( x ) = (1 - -zqx) 9 (zqx)
QX
--e(zx) ~(u) dqu
0
(i -- zu)tD(zu) dqu. (2l)
It is obvious that the resolvent of operator (18) will be conjugate to operator (21). Carry- ing out the necessary calculations, we get
9 ~ (x)
( R ~ Q o ~ ) ( x ) - 1 - - x z
1
i ~ (u) q) (zu) dqu dq ( 1 ]
,~, dqx L (1 --xz) (]:~(zx) t '
q x
from which formula (15) follows immediately. Theorem 3 is proved.
Remark 2. Generalized moment representations of basis hypergeometric series of type (i0) or (13) with continuous measures are constructed in [4]. In the same place, formulas are obtained for their Pade approximant in terms of special biorthogonal systems.
LITERATURE CITED
1. V . K . Dzyadyk, "A generalization of moment problem," Dokl. Akad. Nauk Ukr. SSR, Ser.
A, No. 6, 8-12 (1981).
2. F . H . Jackson, "Transformation of q-series," Messenger Math, 39, 145-153 (1910).
3. G . E . Andrews and R. Askey, "Classical orthogonal polynomials," Lect. Notes Math., 1171, 36-62 (1985).
4. A . P . Golub, "Generalized moment representations and rational approximations," Preprint, Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, No. 87.25 (1987).
5. V . K . Dzyadyk and A. P. Golub, "The generalized moment problem and Pade approximation,"
Preprint, Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, No. 81.58
(1981).
6. R. Walliser, "Rationale Approximation des q-Analogons der Exponential funktion und Ir- rationalitatsaussagen fur diese Funktion," Arch. Math., 44, No. i, 59-64 (1985).
7. R. Askey, "The q-gamma and q-beta functions," Appl. Anal., 8, No. 2, 125-141 (1978).
8. A . F . Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable [in Russian], Nauka, Moscow (1985).