A R K I V F O R M A T E M A T I K B a n d 2 nr 17 Communicated 27 February 1952 by HARALD CRA1K~R and OSKAI~ KLEIN"
Inclusion relations among methods of sulnmability com- pounded from given matrix methods
B y R A L P H P A L M E R A G N E W
1. I n t r o d u c t i o n . I t is our object to clarify and generalize Some theorems on inclusion a m o n g m a t r i x m e t h o d s of s u m m a b i l i t y given b y RUDBERG [1944], a n d to give applications involving t h e Ceshro, Abel, Euler, Borel, binary, a n d o t h e r methods. While RUDBERG gives no references, we observe t h a t some f u n d a m e n t a l ideas u n d e r l y i n g the paper of RUDBER~ and this one were used b y HARDY a n d CHAPMAN [1911], JACOBSTttAL [1920] a n d KNOm" [1920]. Other references a p p e a r later.
F o r each r = 1, 2, 3, . . . , let A (r) be a t r i a n g u l a r m a t r i x of elements ank (r) such t h a t
(1.01) a=k(r) > 0 , a ~ = > 0 , 0 < k < n ; n = 0 , 1 . . . .
(1.02) lim an~ (r) = 0
a n d
n
(1.03) ,~lim k=-~o a~e (r) = 1.
k = 0, 1, 2 . . . .
Then, for each r, A (r) determines a regular Silverman-Toeplitz t r a n s f o r m a t i o n
(1.1) a~(r) = ~ a ~ ( r ) s ~
k = 0
b y means of which a given sequence s~ is evaluable to s if a~ (r) --> s as n -+ c~.
Our terminology agrees with t h a t of HARDY [1949].
L e t t h e elements of a given sequence So, sl, s2 . . . . be denoted b y so(0 ), s 1 (0), s 2 (0) . . . L e t s o (1), sl (1), s 2 (1) . . . . denote the A (1) t r a n s f o r m of s o (0), s 1 (0), s2 (0) . . . . ; let s o (2), s 1 (2), s2 (2) . . . . denote the A 2 t r a n s f o r m of s o (1), s 1 (1), s2(1) . . . . ; a n d so on. Then, for each r = l , 2, 3 . . . .
(1.2) s~ (r) = k~o a~k (r) sk (r - - 1) n = 0, 1, 2 . . . The elements of these sequences form t h e double ~equence
1~. p. ACr{EW, Inclusion relations among methods of summability
(1.3)
so (0), sl (0), s2 (0), s3 (0) . . . .So(l), s1(1), s2(1), s3 (1) . . . . so (2), s~ (2), s~ (2), s3 (2) . . . . so (3), s~ (3), s2 (3), s~ (3) . . . .
F o r e a c h r = l , 2, 3 . . . t h e e l e m e n t s of t h e r - t h row below t h e first in (1.3) c o n s t i t u t e the t r a n s f o r m of t h e given sequence So, s~ . . . . b y t h e p r o d u c t m a - t r i x B ( r ) d e f i n e d b y
(1.4) B(r) = A(r)A (r-- 1) . . . A ( 2 ) A (1).
Thus, for each r = l , 2, 3 . . . .
(1.5) s~ (r) = ~ b ~ (r) s~
k = 0
n = O , 1,2 . . . . w h e r e t h e n u m b e r s b~k(r) a r e the e l e m e n t s of t h e m a t r i x B(r).
S u p p o s e n o w t h a t s~ is a p a r t i c u l a r sequence e v a l u a b l e to s b y one of t h e m a t r i c e s B ( r ) , s a y B(ro). T h e n t h e r0-th row below t h e f i r s t in (1.3) is con- v e r g e n t t o s, a n d i t follows f r o m r e g u l a r i t y of t h e m a t r i c e s Ar t h a t e a c h lower row is likewise c o n v e r g e n t t o s. This m e a n s , r o u g h l y s p e a k i n g , t h a t s~ (r) is n e a r s w h e n e v e r r > r 0 a n d n is large in c o m p a r i s o n t o r. H e n c e , as is well k n o w n a n d e a s i l y shown, t h e r e is a sequence R~ such t h a t R~--> oo a n d t h e r e l a t i o n
(1.6) lim s~ (r~) -- s
n . o o
holds for each sequence r~ such t h a t r 0 <_- r~ < R~ for e a c h n. This i m p l i e s t h a t if s~ is a sequence e v a l u a b l e t o s b y one of .the m a t r i c e s B (r), t h e n t h e r e is a sequence r0, rl . . . . such t h a t r~ ~ oo a n d t h e sequence s~ is e v a l u a b l e to s b y t h e m a t r i x w h i c h t r a n s f o r m s sn i n t o
b
(1.7) s~ (r~) = k~__0 ~ (r~) sk, n = 0, 1, 2 . . .
W h i l e B (r0, rl . . . . ) a n d bnk (to, rl . . . . ) a r e n a t u r a l n o t a t i o n s for t h i s m a t r i x a n d i t s elements, we see t h a t b,~(ro, r l , . . . ) = b~ (rn). H e n c e we shall use t h e s i m p l e r n o t a t i o n s B (r~) a n d b~k (r~) for t h e m a t r i x a n d its elements. W e o b s e r v e t h a t if r . = r for e a c h n, t h e n B(r~)= B(r). W h i l e i t c o u l d be p r e s u m e d t h a t d i f f e r e n t sequences r~ would be r e q u i r e d for d i f f e r e n t sequences s~, we shall see in S e c t i o n 3 t h a t t h i s is n o t so. I n o t h e r words, a single sequence r~ c a n b e chosen i n such a w a y t h a t t h e m a t r i x B{r~) defines a m e t h o d of s u m m a b i l i t y w h i c h includes e a c h one of t h e m e t h o d s B I , B 2, . . . . Moreover, r e l a t i o n s a m o n g some such m e t h o d s B(r~) will be o b t a i n e d . W e shall s o m e t i m e s refer t o a m a t r i x B ( r ~ ) , w h i c h r e s u l t s f r o m c o m b i n i n g m a t r i c e s A (r) a n d selecting f r o m m a t r i c e s B(r), as a compounded m a t r i x .
ARKIV FOR MATEMATIK. B d 2 nr 17
2. Two examples. Before passing to constructive theorems and significant examples, we look briefly at two trivial examples. Suppose first that A (r) is, for each r = 1, 2, . . . , the identity matrix which transforms each sequence into itself. Then each row of the double sequence (1.3) is identical with the first row.
I t is therefore clear that the transformation B(r~) defined by (1.7) is regular
r p
if and only if r~ -+ o0, and, moreover, that two transformations B(rn) and B ( ~ ) for which rn ~ c~ and r',~ --> c~ are equivalent to convergence and hence to each other.
Our second trivial example involves the double sequence (2.1) 2, s 1(0), 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, . . .
2, s 1(1), 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, . . .
2, 81 (2), 0, 1, 0, 1, 0, 1, 0, 1, 0, 1 . . . . 2, s~(3), 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 . . . . 2, s~(4),0, 1,0, 1,0, 1,0, 1,0, 1 . . . .
in which s 1 (0), sl (1) . . . . is a given sequence of real numbers such t h a t - 2 =
= s l ( 0 ) < s l ( 1 ) < s ~ ( 2 ) < . . . a n d s l ( r ) < - - l f o r e a c h r = 0 , 1 , 2 . . . For each
= 2, 3, 4, . . . the row containing sl(r) is identical with the row containing s l ( r - - 2 ) except that s l ( r ) ~ s 1 ( r - 2 ) . For each r = l , 2, 3, . . . , l e t A(r) b e a matrix which carries the row of (2.1) containing s l ( r - - 1 ) into the row con- taining sl (r) and which is not only regular but satisfies the stronger conditions
(2.2) ank (r) > O,
and
n
(2.3) e=~x ank (r) = 1
0 < k < n ; n = O , 1 , 2 . . . .
n = 0 , 1 , 2 . . . I t is easy to see that such matrices A (r) exist, but are not uniquely determined by our specification of the sequences obtained by starting with the one partic- ular sequence given in the first row of (2.1). The service of this example, to which one may profitably refer occasionally, is to establish the t r u t h of asser- tions such as the following. The hypotheses on A (r) given in Section 1, even when supplemented by the stronger hypotheses in (2.2)and (2.3), imply neither
r ~
equivalence nor consistency of the two transformations B ( r ~ ) a n d B ( n ) of the form (1.7) for which r~ and r~ are the sequences 1, 2, 1, 2, 1, 2 . . . . and 2, 3~ 2, 3, 2, 3, . . . .
3. I n c l u s i o n t h e o r e m s . The theorems of this section are generalizations of Theorem 1 of RUDBERG [1944]. The matrices/'1, T~ . . . . introduced at the top of page 2 of Rudberg's paper correspond to the matrices A(1), A(2), . . . of this paper. Rudberg assumed, in the notation of this paper, t h a t ank > 0 when
n
0 < k = < n ; and he needed the c o n d i t i o n k ~ 0 a ~ k ( r ) = l , n = 0 , 1, 2 , . . . , to obtain some of his equalities, i t should be pointed out t h a t the matrices A (r)
R. P. AGNEW, Inclusion relations among methods of summability
a n d B ( r ) , which are d i f f e r e n t e x c e p t in v e r y special cases, are d e n o t e d b y t h e s a m e s y m b o l TT in R u d b e r g ' s p a p e r ; a c c o r d i n g l y one c a n n o t i n t e r p r e t t h e a r g u - m e n t s a n d r e s u l t s of t h a t p a p e r , w i t h o u t g i v i n g v e r y close a t t e n t i o n t o t h e details.
, T h e o r e m 3.1. There is a monotone increasing sequence R1, R2, . . . such that R~ > 1, R~ ---> c~, and the matrix trans]ormation B(rn) de/ined by (1.7) is regular whenever 1 < r~ < Rn ~or each n.
P r o o f . Since t h e m a t r i x B(r), d e f i n e d b y (1.4), is t h e p r o d u c t of r e g u l a r m a t r i c e s w i t h n o n n e g a t i v e e l e m e n t s , i t h a s t h e s e s a m e p r o p e r t i e s . L e t
R R
(3.11) F ( R , n ) = ~ o T~I bnk(r) R = 1, 2 . . .
T h e n (3.12) H e n c e
I f 1 < rn < R~, t h e n (3.13)
L e t (3.14) T h e n
(3.15)
!imr162 F (R, n) = 0 R = 1, 2 . . . t h e r e is a sequence R ) such t h a t R' n -+ oo a n d F (R'~, n) ---> 0 as n -+ oo.
lim b~, (r~) = 0 k = O, 1, 2 . . .
n §
lira G(R, n) = O R = l, 2, . ....
H e n c e t h e r e is a sequence R " such t h a t R " n -+ c~ a n d G (R'~', n) ---> O. I f 1 < =
< rn < R " t h e n
.
(3.16) lira b~k (rn) 1.
If, for each n, R~ is t h e m i n i m u m of t h e p o s i t i v e i n t e g e r s R'~, R'~', R' n + l ~ R " n + l ~
R ~ + 2 , . . . t h e n 1 < R 1 < R 2 < . . . , R n - - > c ~ , a n d b o t h (3.13) a n d (3.16) h o l d w h e n 1 < r~ < R~. Since b~k(r~) > 0, i t follows t h a t bnk(rn) is r e g u l a r when 1 < r~ _-< R~ a n d T h e o r e m 3.1 is p r o v e d .
T h e o r e m 3.2. I] r'~ and rn are sequences such that r~ is monotone increasing, i~ r; > r~ /or all su//iciently great n, and i/ the trans/ormations B(r~) and B(rn)
r I
are regular, then B ( ~ ) includes B (r~).
P r o o f . Since t h e a l t e r a t i o n of a f i n i t e set of e l e m e n t s of t h e sequence r~
h a s no effect on B(r'n) e v a l u a b i l i t y of sequences, we can a n d shall a s s u m e t h a t r~ _-> r~ for each n = 1, 2, 3, . . . . L e t n be a f i x e d p o s i t i v e integer. W e m a k e t e m p o r a r y use of t h e a b b r e v i a t i o n a = L [ql, q2 . . . q v ] t o signify t h a t a is a l i n e a r c o m b i n a t i o n , w i t h n o n n e g a t i v e m u l t i p l i e r s , of t h e n u m b e r s in b r a c k e t s . Since r~ > r~, our h y p o t h e s e s on t h e m a t r i c e s A (r) i m p l y t h a t
A R K I V F O R M A T E M A T I K , Bd 2 n r 17 (3.21) s,(r'~) = L [ s o ( r ' ~ - - l ) , s l ( r ' - - 1 ) , . . . , sn(r~ - - t)] . . . .
= L[so(r~), s l ( r ~ ) , . . . , s=( ~)j r
= L[so(r=), . . . , s ~ - l ( r ~ ) ] + c~s,~(r~)
where c , , > 0. Since rk > rk-1, this reduction can be continued until we obtain nonnegative coefficients c~k such t h a t
. n
(3.22) s~(r~) = k ~ o C ~ S k ( r k ) n = 1, 2, 3 . . .
Since c ~ -> 0, we can show t h a t this t r a n s f o r m a t i o n is regular, a n d hence t h a t B ( r ' ) includes B(r~), by showing t h a t
(3.23) lira e~k = 0 k = 0, 1, 2 . . .
and
n
(3.24) lim k~o c~k = 1.
To prove (3.23), we use (3.22) a n d (1.7) to obtain
n
(3.25) s~(~) = ;-~1 c~sj(rs)
n ]
Since the m a t r i x B (r~) which transforms s~ into sn ( r ' ) is regular b y hypothesis, it m u s t be true t h a t
n
(3.251) lim ~ c~jbj~(r~) = O, k = O, 1, 2 . . .
Since all terms a p p e a r i n g in these sums are nonnegative, it follows t h a t
( 3 . 2 5 2 ) = 0, k = 0, 1, 2 . . .
Since the hypothesis t h a t akk ( r ) > 0 for each k a n d r implies t h a t bkk ( r ) > 0 for each k a n d r, t h e conclusion (3.23) follows from (3.252). To prove (3.24), we observe from (1.7) t h a t s~(r~) a n d s~(r'~) are the B ( r n ) a n d B(r'~) trans- forms of t h e sequence sn for which sn = 1 when n = 0, ], 2 . . . Since B (r=)
r I
a n d B ( ~ ) are regular b y hypothesis, it follows t h a t s=(r=)= 1 + s = a n d
r p
s n ( r ' ) = 1 + e ~ where ~ - + 0 a n d s ~ - + 0 as n - > c ~ . Use of (3.22) then gives (3.26)
n
1 + s~ = k=~o c~k(1 + ~k).
R. P. AGNEW, Inclusio~ relations among methods of summability
Since c~k > 0 and (3.23) holds, this implies (3.24) and Theorem 3.2 is proved.
The fact t h a t the conclusion of Theorem 3.2 will fail to be obtainable if we delete one or the other of the hypotheses t h a t B ( r ; ) a n d B ( r n ) a r e regular can be seen from the example of the unique sequence of matrices A (r) for which the double sequence (1.3) takes the form
(3.27) So, sl, s2, s3, s~, s s , . . . So, 281, s2, s3, st, Ss, . . .
80~ Sl~ S2~ 83~ 84~ 8 5 , 9 . . SO~ Sl~ S 2 , 2s3, s4, ss, . . .
Here sn ( r ) = s~ except t h a t s t ( r ) = 2st when r is odd. Another example is ob- tained by making s t ( r ) = 2st for each r > 0. if, however, the matrices A(r) satisfy the additional condition
n
(3.28) ~ oa~k(r) = 1 n = 0, 1, ~) -, . . .
then the matrices B(r) satisfy the condition
(3.281) ~ b ~ k ( r ) = 1 n=0.~ 1,-,9 . . . ;
k = 0
in this case the numbers G and sk in (3.26) are zero and we obtain the con- clusion of Theorem (3.2) without the hypothesis t h a t B(rn) is regular.
T h e o r e m 3.3. There is a monotone increasing sequence R1, R~ . . . . such that Rn >= 1, Rn -~ c<), and the matrix trans/ormation B (rn) includes each one o / B (1), B(2), B(3) . . . . whenever 1 <= r~ < Rn and r n -~ C~.
P r o o f . With the sequence R1, R2 . . . . determined as in Theorem 3.1, we obtain the desired conclusion with the aid of Theorem 3.2.
T h e o r e m 3.4. The regular trans/ormations o~ the /orm B (r~) [or which rl <--
< r2 < ra <= . . . constitute a consistent /amily.
P r o o f . I,et r~ and r'~ denote monotone increasing sequences of integers such t h a t B(r'~) and B ( r ' ) are regular. F o r each n, let r~ be the m a x i m u m of r~
and r " . Then, for each n, one or the other of the two formulas
r r
(3.41) a n k ( r , ) = a~k(r'), a~k(r~) = a~k( ~)
holds when 0 < k < n. Thus regularity of B ( G ) and B ( G ' ) implies t h a t of B(rn).
Hence Theorem 3.2 implies t h a t B(r~) includes both B ( r ' ) and B ( G ' ) . There- fore B(r'~) and BIr"~ ~ t must be consistent and Theorem 3.4 is proved.
4. The condition a ~ k ( r ) > O. We assumed in (1.01) t h a t a~k(r)>= 0 for each n, k. and r. If this hypothesis were deleted and replaced by the hypothesis 366
/kRKIV FOR MATEMATIK. Bd 2 nr 17
k = 0
t h e n the matrices
A(r)
would still be regular, b u t Theorem 3.1 and our de- ductions from it would fail. The following example provides proof. L e t T be the regu- lar transformation which transforms s~ into So, 2 s o - - s~, 2 s~ - - s2, 2 s2 - s3 . . . L e t A ( r ) = T for each r = 1 , 2 . . . . so t h a t B ( r ) = T ~. I t is readily verified that, for this example,B(r,)
is regular if and only if the sequence rn is bounded.5. Applicability o f T h e o r e m 3.4. Theorem 3.4 establishes existence of m a t r i x methods
B(r=)
of summability such t h a tB(r~)
includesB(r)for
each r == 1, 2, 3 . . . This does n o t necessarily imply t h a t B (r~) includes A (r) for each r = 1, 2, . . . . Suppose, for example, t h a t A~ and A2 are two inconsistent methods. Then no m e t h o d can include b o t h A~ and A~. The point is, of course, t h a t the relations
B~A~
a n dB ~ A 2 A ~
do n o t imply t h a t B ~ A 2.I t therefore becomes of particular interest to know w h a t given sequences of m a t r i x methods are representable in t h e form B(1), B(2), . . . defined in Section 1. The answer is obvious. A given sequence B ( ] ) , / } ( 2 ) . . . . has the form if and only if (i) for each r = 1, 2, 3 . . . . the m a t r i x B (r) is a regular triangular m a t r i x of nonnegative elements which has an inverse and
(i i)the
matrices .4 (1), zi ( 2 ) . . . defined b y A ( 1 ) = / ~ ( 1 ) and(5.1) A ( r ) = B ( r ) B - ~ ( r - - 1) r = 2, 3 , . . .
are such that, for each r = 1, 2, . . . , A ( r ) is a regular triangular m a t r i x of nonnegative elements which has an inverse. W h e n a given sequence B (r) has t h e properties (i) and
(i i),
p u t t i n g A ( r ) = z] (r) givesB(r)= B(r).
There is an i m p o r t a n t case in which the hypothesis t h a t
(5.2)
B(r~)~A(r)A(r--1)
A(1), r 1, '}implies t h a t
B(rn)~A(r)
for each r = 1, 2 , . . . . Suppose the matricesA(r),
r = 1, 2 , . . . satisfy the conditions of Section 1 and, in addition, constitute a c o m m u t a t i v e family in t h e sense t h a t
A(r)A (s)= A (s) A (r) for
each r, s == l, 2 . . . Then
(5.3)
.4(r)A (r--
1) . . . . 4 (1) =.4(1)A(2) . . . A(r)mA(r)
and accordingly (5.2) implies t h a t
B(r~)~A (r)
for each r = 1, 2 . . .The Cess and Euler matrices are members of a large family of c o m n m t a t i v e matrices first studied b y HURWITZ and SILVERMAN [1917] a n d b y HAUSDORF~
[1921]..
6. Ceshro m e t h o d s . F o r each positive n u m b e r :r let C (a) denote the Ces~ro m a t r i x of order r which t r a n s f o r m s a given sequence sn into
k=0 n - - I t / n 8k.
R. P. AGNEW, Inclusion relations among methods of summability
If ~1, ~2 . . . . is a monotone increasing sequence of positive numbers, then for each fixed r, the matrices C(~r) and C(~r)C l(~r_~) are regular triangular matrices of nonnegative elements which have inverses; see HAUSDORFF [1921].
Hence our theorems apply to the case in which the matrices B ( r ) are the Ces~ro matrices C ( ~ ) . i t is well known and easy to show that, since the elements of Ces~ro matrices of positive order are positive and satisfy the strong condition in (3.281), the compounded Cess m a t r i x C(~n) is regular if and only if its elements C,k(~n) are such t h a t lira C = k ( ~ n ) = 0 for each k and hence if
n --~ r162
and only if lira :~(n)/n = 0. The nonregular m a t r i x C(~,) for which ~n = n, and a closely related regular m a t r i x t h a t includes C(~) for each fixed positive
~, has been studied by OBRECHKOFF [1926]; see also KOGBETLIANTZ [1931, page 47]. RU~)BERG [1944, Th6or6me I I I ] compared methods of the form C ( ~ n ) w i t h the Abel power series method. We turn to this subject here because the stated results are supported by inadequate arguments and some are false.
A series E u , of complex terms is evaluable to s by the A b e l po.wer series method P if the series in
(6.02) P ( x ) = ~ x k u k
k = 0
converges when 0 < x < l to a function P ( x ) such t h a t P ( x ) - ~ s as x ~ l . T h e o r e m 6 . t . I~ 2 u ~ is a series such that Y~x kuk converges when 0 < x
< 1, and i] xl, x 2 . . . . is a sequence such that 0 < xn < 1 and x~ ---> l, then there is a regular compounded Cesdro matrix C(a=) such that
(6.11) lim I (rn (~n) - - P (xv(n)) [ = 0
n - - ~
where p(1), p(2) . . . . is a monotone increasing sequence o/ positive integers which contains each positive integer and /or which p (n)-+ c~ as n - ~ c~.
L e t Eu= be a given series satisfying the hypotheses of the theorem. F o r each q = 1, 2, 3, . . . let k(q) be the least integer greater than q such t h a t
(6.12) ~ I x ~ u k [ < l .
k = k (q) q
The series-to-sequence form of the Cess transformation (6.01) is
~ U k
k=0~ n - - k n The coefficient of uk in (6.13) is
n ( n - - 1 ) ( n - - 2 ) . . . ( n - - k + l) (6.14) c.k (o~) =
( n + ~ ) ( n + ~ - - 1 ) ( n + ~ - - 2 ) . . ( n + ~ - - k + 1) Setting x = (1 + ~/'n) 2, we p u t this in the form
1 - 1 1 2 1 - k - - 1
(6.15) c.k (~) n n n xk"
1 - - x 1 --2~x 1 ( k - - 1)x
n n n
368
A R K I V F O R M A T E M A T I K . Bd 2 n r 1 7
With positive integers p(1), p(2) . . . . to be determined below, the equivalent formulas
(6.16) ~ = n x--~(,)--I , x,<~) = 1 1 +
we define con by
and put the C(a~) transform of Y,u~ in the form
o ~
(6.17) a~(~n) ~ ~."n k io (n) U/c x ~
k = 0
where ~ 0 = 1,
(6.18) Ynk =
1 2 k - - 1
1 - - - 1 - - - 1 - - - -
n n n
xv <~) 2 xp (n) (k - - 1) xv <n)
1 1 1
n n n
when 0 ~ k < n, and y~k = 0 when k ~ n. Since 0 < ynk < 1, we find that (6.2)
c~
I ~. ( : : ) - P(..,.~)I : ~ o l ~ - 1[ * ~,`") l u~ I
k (p~n)) X k oo
k = O k f k ( l ~ ( n ) )
Xp(n) ] Uk l * k
Since (6.12) shows that the last term of (6.2) is less than 1 / p ( n ) , we can obtain the desired conclusion (6.11) by determining a sequence p ( n ) of the required type such t h a t
k (p(n)) 1
(6.21) kffiO , . v I rn~ - a J ~ . < . ) l u k I < p(n) - -
for each sufficiently great n. Using (6.18), we see t h a t we can choose an integer n 1 ~ 1 such t h a t (6.21) holds when p ( n ) = 1 and n > nl. Then choose n 2 ~ % such that (6.21) holds when p (n) = 2 and n > n 2. Continue the process to obtain an increasing sequence nj of integers such t h a t (6.21) holds when p ( n ) = / " and n _ - n j . On setting p(n) = 1 when l < n ~ n 2 , p ( n ) = 2 w h e n n 2 = < - n ~ n a , and so on, we obtain the sequence p i n) and complete the proof of Theorem 6.1.
The impossibility of proving (6.11) with p ( n ) = n follows from consideration of the series Z u n for which un = 1, sn = n + 1 and P ( x ) = 1 ] ( 1 - - x ) . I n this case an(a,) always lies between 0 and n + 1 while P ( x n ) could be (n + 1) 2.
T h e o r e m 6.3. / / Z un is a series evaluable to s by the Abel power series method P, then there is a regular compounded" Cesdro method C(o~n) by which the series is also evaluable to s.
This follows immediately from Theorem 6.1, since (6.11) and the consequence lira P(x~<n)) = s of Abel evaluability imply t h a t lim a~ (~,) = s. The same argu-
n. P. AGr~EW, Inclusion relations among methods of summability
ment shows that if the Abel transform P ( x ) exists over 0 < x < 1 but, as is true in the case of the example P ( x ) = sin ( 1/(1 x)}, different sequences xn give different limit points lira P ( x n ) of P ( x ) , then each limit point must be the actual limit of ~n(~n) for some regular compounded Ces~ro matrix C(~n).
Such a compounded Ces~ro matrix generates a method of summability not included by the Abel method, and two such matrices can generate inconsistent methods which, by Theorem 3.4, cannot both have the form C(~n) where ~ is monotone increasing.
7. Euler methods. For each ~ for which 0 < ~ < 1, let E (~) denote the Euler matrix of order ~ which transforms a given sequence s, into
(7.0~) E~(~) = ~ : 0 ~ k !
If ~,, ~2 . . . . is a monotone decreasing sequence of positive numbers for which
~1 < 1 then, for each r the matrices E ( ~ ) and E ( : c ~ ) E - * ( u ~ _ I ) a r e regular triangular matrices which have inverses and nonnegative elements; for these and other facts relating to Euler transformations, see AGNEW [1944], references given there, and HARDY [1949]. Hence the theorems of Section 3 apply to cases in which the matrices B ( r ) are the matrices E(~.). A compounded Euler matrix E(~n) with elements enk(an) is regular if and only if lira enk(~n)= 0
n :-> g o
for each k and hence if and only if lira nun = c<~. RUDBERG [1944, Th6or~me IV] compared methods E ( 1 / n ) with the Borel exponential method B*.
A series Z un with partial sums sn is evaluable to s* by the Borel exponential method B* if the series in
z X
(7.02) a* (x) = e- ~0: k.v s~
converges for each x > 0 and a* (x)-+ s* as x - + c~.
T h e o r e m 7.1. I / so, s 1 . . . . is a sequence such that E ( x k / k 1)sk converges ]or each x > O and i] xl , x2 , . . . is a sequence such that xn > O and xn --+ o% then there is a regular compounded Euler matrix E(~n) such that
(7.11)
l i r a I E .(~n)
- (r* (X,<n)) I = 0n .--~ g o
where p(1), p(2) . . . . is a monotone increasing sequence o] positive integers which contains each positive integer and ]or which p ( n ) ~ go as n - ~ c~.
Let So, s,, . . . be a given sequence satisfying the hypotheses of the theorem.
For each q = 1, 2, 3 . . . . let k(q) be the least integer greater than q such t h a t
(7.12) I
k = ~ <q> q
Choose n, such t h a t xl < n , and x~ < nl. Let ~n = p(n) = 1 when 1 < n _-< hi.
With positive integers p (n, + 1), p (hi +7 2) . . . to be determined below to satisfy the conditions of the theorem and the additional condition
ARKIV FbR MATEMATIK. B d 2 nr 17
(7.13) xp(~)/n < 1,
we define ~ when n > n~ b y t h e e q u i v a l e n t f o r n m l a s
n _--> n l ,
(7.14) ~= = x,~n)/n, xp(n) = n ~
a n d p u t t h e E ( ~ ) t r a n s f o r m of sn in t h e form
k•=
Xp (n)(7.15) E , (an) = an k (1 - - a = ) " - k S k = ~V~k k!-
0 k = 0
w h e r e
(7.16) ~f=k : (n ~ k)! n ~
Sk
w h e n O _ - < k - < n a n d ~ v . k = 0 w h e n k > n . Since 0 < y ~ z _ < - 1 a n d O _ - _ e x p - 9 [ - xv(~)] =< 1, we f i n d t h a t w h e n n > n~
k (p (n)) X k
(7.17)
IE,~(~Zn--a*(Xp(n))l <=y_, IWnk--e- ,(n) I "(n) v(n)
+ ~ xk
k=k(p(n)) k! ISk]"
Since (7.12) shows t h a t t h e l a s t t e r m of (7.17) is less t h a n 1 / ( p - - n ) , we cart
" o b t a i n t h e d e s i r e d conclusion (7.11) b y d e t e r m i n i n g a sequence p ( n ) o f t h e r e q u i r e d t y p e such t h a t
k (p (n)) /c 1
(7.18) Z
- - e - , ( " ) I - - - [ s k i <.x,(n)
--k=o lc ! p (n)
for e a c h s u f f i c i e n t l y g r e a t n. I t f o l l o w s . f r o m (7.16) t h a t if p(n) has a c o n s t a n t v a l u e q, t h e n (7.18) will h o l d for all s u f f i c i e n t l y g r e a t v a l u e s of n. W e h a v e a l r e a d y d e f i n e d n l . W h e n j > 1 a n d nj_ 1 has b e e n defined, choose nj such t h a t n j _ ~ < n j , x k < n j w h e n k = 1, 2 . . . . , i, a n d such t h a t (7.13) h o l d s w h e n n > nj a n d p ( n ) = j - - 1 . I n t e r m s of n l , n~, n 3 . . . . we now c o m p l e t e t h e defini- t i o n of t h e sequence p (n) b y s e t t i n g p (n) = 1 w h e n nl < n < n~ and, for e a c h i = 2, 3, : . . , p(n) = j - - 1 w h e n nj < n < n j + l . T h e n p(n) = 1 w h e n 1 < n < n 3 a n d (7.18) h o l d s w h e n n _-> n2. To show t h a t (7.13) holds, we o b s e r v e t h a t if nl < n < m . t h e n x v ( , ) = x l < n l ~ n a n d if j > 2 a n d ni < n < n j + l t h e n (7.19) x v ( n ) = x j - 1 --< 7/;/-1 < n / <: n .
I t is now o b v i o u s t h a t t h e sequences p (n) a n d ~ h a v e all of t h e r e q u i r e d p r o p - erties a n d T h e o r e m 7.1 is p r o v e d .
T h e o r e m 7.2. I[ Z u , is a series with partial sums s, which is evaluable to s by the Borel exponential method B*, then there is a regular compounded Euler method E(ctn) by which the series is also evaluable to s.
371
R. P. AGI~EW, Inclusion relations among methods of summability
This follows from Theorem 7.1 in the same w a y t h a t Theorem 6.3 follows f r o m Theorem 6.1. Obvious modifications of the remarks following Theorem 6.3 a p p l y here.
8. T h e b i n a r y , E u l e r , a n d Borel m e t h o d s . Corresponding to each :r in the interval 0 < ~ < 1, let T(:r denote the m a t r i x of the b i n a r y transformation of order ~ which transforms a given sequence So, s~, s2, . . . into the sequence
(8.1)
so, (1 - - z r o + ~81, (1 - - ~ ) s 1 + ~s e . . .The t r a n s f o r m a t i o n s T ( a ) were used b y HuRwITz [1926] to illustrate the t h e o r y of Tauberian theorems. The special t r a n s f o r m a t i o n T ( 1 / 2 ) and its powers are a m o n g those studied b y SILVERMAN a n d Szasz [1944] a n d b y SZASZ [1944]. The matrices T (:r satisfy the conditions imposed upon the matrices A (r) in Section l. H e n c e we m a y set A (r) = T (:r for each r = l, 2 . . . . a n d obtain B (r) = T~(:r for each r = l, 2, . . . .
F o r each r = 0 , 1 , 2 , . . . , l e t the t r a n s f o r m of a given sequence s~ by the m a t r i x T~(:r be denoted b y
(8.2)
s o (r, :r sl (r, ~), s 2 (r, ~) . . . . ;in particular, s~ (0, ~) = s~. To facilitate the writing of formulas, let s~ (r, :r be defined for negative integer values of n b y the formulas
(8.3)
s n ( r , or = s o(r, ~r = s o , n = - - 1 , - - 2 , - - 3 . . . Then, for each n = 0, • 1, q- 2 . . . . ,s~ (1, :r = (1 - - ~r 1 + 0~s, (8.41)
a n d
(8.42) s,(2; ~) = ( 1 - - ~ ) s , _ i ( 1 , ~) + a s , 0 , ~)
= ( 1 - - a ) 2 s , _ 2 + 2 a ( 1 - - : r + :r Thus when r = 1 a n d when r = 2 the formula
(8.43) s , ( r , ~ ) = k = n _ r ~ ( n - - k r ) r + k _ n ( I _ C r ~
holds when n = 0, + 1, • 2 , . . . ; and it is easily shown b y induction t h a t (8.43) holds for each r = 1, 2, 3 . . . The double sequence whose rows consist of the t r a n s f o r m s of s~ b y the various powers of T (~) t u r n o u t t o be v e r y inter- esting. I n particular, the sequence of elements on t h e m a i n diagonal of t h e double sequence is generated b y the regular t r a n s f o r m a t i o n b y w h i c h s~ is evaluable to s if sn (n, a ) - + s where
(8.5)
n ( n ) : c k ( 1 - - ~ ) ' - ~ S k ;s~(n, :r k
A R K I V F O R M A T E M A T I K . B d 2 n r 1 7
and (8.5) is precisely the Euler transformation E (a) of order ~. I t follows from Theorem 3.2 t h a t T ( a ) and all of its powers are included by E(~). Using the v e r y well known fact (see HARDY [1949, p. 183]) t h a t E (a) is included by B*, we conclude t h a t T(~) and all of its powers are included b y B*.
Therefore the extensive family F of regular transformations, whose elements comprise the totality of binary transformations T(~) for which 0 < ~ < 1 and the totality of positive integer powers of these, constitutes a consistent family F of transformations included b y the Borel exponential method B*.
9. The binary, Niirlund, and generalized Abel methods. Let T (~) be the binary transformation defined in Section 8. Each sequence Po, Pl, P2 . . . . for which p > 0, p= =>_ 0, and p = / P ~ ~ 0 where P~ = Po + Pl A- "'" -4- Pn, generates a NSrlund transformation N (pn) by which a sequence s= is evaluable to s if a= -+ s where (9.1) ~ = ( p n s o + p ~ _ l s 1 + . . . + p o S ~ ) / P ~ , n = O, 1 . . .
:For each ~ = 1, 2, 3 . . . . the transform s= (r, ~) of a sequence s~ by the trans- formation T r (~) is almost identical with the transform an (r, ~) by the N6rlund transformation N ( p = ( r , ~)) generated b y the sequence po(r, o~), p l ( r , ~) . . . . for which
/ \
(9.2)
and p = ( r , ~ ) = 0 when n > r ; in fact (8.43) shows t h a t
(9.3) s~ (r, ~ ) = ~= (r, ~) n -> r.
I t follows from this t h a t T (r) (:r is equivalent to N ( p = (r, ~)), and using results of Section 8 we see t h a t such a N5rlund method is included by the Euler method E ( r ) an d hence is also included b y the Bore] exponential method B*.
L e t a series Y u~ with partial sums s= be called evaluable to a by the gener- alized Abel method P* if the series in
or
(9.4) / (z) = (1 - - Z)k~oZk SZ
has a positive radius of convergence R and the function / ( z ) defined by (9.4) when [z I < R determines, by analytic extension along radial lines emanating from the origin, a function / (z) such t h a t / (z) -+ a as z -+ 1 over the real inter- val 0 < z < 1. I t was shown by SILVERMAN and TAMARKIN [1929] and by TA-
~ n a K i ~ [1932] t h a t if N (p,) is regular and ~, > 0 then N (p=) c P* ; and the same proof shows t h a t if N ( p ~ ) is regular, p 0 > 0 , and p , > 0 , then N ( p , ) c P * . Therefore T(~) and all of its powers are included in P*. This shows again t h a t the family F, consisting of the binary transformations T (~) for which 0 < ~ < 1 a n d their powers, constitute a consistent family.
I t is not true t h a t the ordinary Abel method P includes all members of F.
F o r example, it is easy to show t h a t the sequence s~ = ( - - 2 ) " is evaluable T (2/3) to 0; but in this case the series E z ~ s= has radius of convergence 1/2 and therefore the sequence is nonevaluable by the Abel method P.
10. The symmetric binary method and its powers. Let T denote the symmetric binary method T ( 1 / 2 ) which transforms a given sequence s o, s~, s2, . . . into
R. P..a_GNEW, Inclusion relations among methods of summability
(10.1)
So, 2 (So + sl), 2 1 1 (81 • 82), 2 (82 ~- 83), . 1 ..SzAsz [1944] used this simple method T to illustrate m a n y points in the theory of summability. In particular, he found it very easy to show t h a t for each r = 0, l, 2, . . . the alternating zeta series, the series in the right member of the familiar equation
1 1 1 1
(10.2) ( 1 - 2 ~ - 1 ) ~ ( s ) - is 22 + 3 ~ - 4 - ~ + -..
is evaluable T T for each s = a + it in the half-plane a > - - r . This simple result and Sections 8 and 9 show t h a t the alternating zeta series is evaluable over the entire plane by the Euler method E ( 1 / 2 ) and,hence also by the Borel exponential method B* and the generalized Abel power series method P*.
White Hall, Cornell University, Ithaca, New York, USA.
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a n d S z a s z , O. 1944 O n a class of NSrlund matrices. Annals of M a t h e m a t i c s (2) vol. 45, pp.
347-357. - - S i l v e r m a n , L . L . a n d T a m a r k i n , J . D. 1"929 O n t h e generalization of Abel's the- o r e m for certain definitions of s u m m a b i l i t y . M a t h e m a t i s c h e Zeitschrift, vol. 29, pp. 161-170. - - S z a s z , O t t o . 1944 I n t r o d u c t i o n to t h e t h e o r y of d i v e r g e n t series. D e p a r t m e n t of Mathematics, U n i v e r s i t y of Cincinnati. 72 pp. - - T a m a r k i n , J . D. 1932 R e m a r k s of t h e t r a n s l a t o r . (On N S r l u n d s u m m a b i l i t y ; a p p e n d e d to the t r a n s l a t i o n of a note b y Woronoi.) A n n a l s of M a t h e m a t i c s (2) vol. 33, pp. 423-428.
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Tryckt den 17 september 1952 Uppsala 1952. Almqvist & Wiksells Boktryckeri AB
