Pairs oï matrices and unbounded sequences

R O C Z N IK I P O L S K IE G O T O W A R Z Y S T W A M A T EM A T Y C Z N EG O Séria I : P R A C E M A T EM A T Y C Z N E X X I I I (1983)

G. M.

(Christchurch, Hew Zealand)

Pairs oï matrices and unbounded sequences

1. Let A = (amn) be a regular limitation matrix and f{ n ) ,f( n ) / oo a monotonically increasing function. Then A (f( n )) is defined as the subset of

bV l*J ^ Щ ( п) (n = b 2 , ...) for some M > 0}

limited by i .

For the set of bounded sequences limited by A we shall write A.

By A (f( n)) + В (/(п )) we denote sequences of the form {oon-\-yn} where {#„} e A (f(n)) and {yn} e B (f(n)). If A(f(n)) + B (f(n )) includes all bounded sequences, then A (f(n)) -\-B{f(n)) is said to span the bounded sequences.

The value to which s is limited by A = (am>n) is denoted by JL-lims; A trans­

forms the sequence s into the sequence {Am} (or (Am(sn)}), i.e.

- 1 ^m,n^n ■

Two regular matrices A = (amn) and В = (bmn) are said to be /(^ -c o n ­ sistent (b-consistent) if s e A [f(n ))n B [f(n ))(s e A n B ) implies that

J.-lim s = B-lim s.

The following has already been proved (see [4], [5]):

Th e o r e m

1. I f A +

spans the bounded sequences, then A and В are not b-consistent.

The object of this paper is to extend this result so as to show that if 4 (/(»)) + B (f(n )) spans the bounded sequences, then A and В are not f(n )- consistent. This theorem is not proved for all pairs of regular matrices but for a large subclass of regular matrices.

2. For any regular matrix A = (am>n) and f( n ) / oo it is easy to show the existence of X (m )/ oo such that A(m) — Я (т —1 ) < 1 (m = 2 , 3 , . . . ) and

Цт)

lim £ f ( n ) \ a m>n\ = 0 .

rn->CO n = 1

( 1 )

D

1. Let {A1} (1 < i < N) be a finite set of regular matrices, am,n = 0, n

y{m)

i

N), fi(m)

and X{m) a function such that

is satisfied for all of the matrices in the set.

The matrices {A1} have an f(n ) singularity, 8{f(n)), if for every s > 0 there exist K (e) and r(s) such that for every к > r(s) there exist sequences

{ocQ (1 < N) satisfying:

( 2 ) (3)

and (4)

We now prove the following:

T

2. Let {A1} (1 < i < N), агтгП ~ 0, n ^ y{m) be a finite set o f regular matrices which do not have an f(n ) singularity. Then there exists a regular matrix A — (am>n) such that

A (f(n )) = ^ A*(f(n))-

i —l

P ro o f. Without loss of generality we shall assume that агт>п — 0 {n < X(m)) and proceed to construct the matrix A, see [1] or [4], pp. 11C, 117.

Since there is not an f{n ) singularity, there exists an e > 0 such that for every r and К there exists а к = к (г) > r such that (2 ), (3) and (4) cannot hold simultaneously. For each integer m, let r — X(m) and consider the vector space B {r, k) =

= 0, n < r, n > k}. Let

N

X = {а?г'; 1 < г < N), f r(X) = sup \x{n\ /X f{n), r^n^k г-=1

and

N

gr{X) = sup £ \A\{4)|

г = 1

wlioro fill о sup is liukori over tliosc t sxLcb. tliriti

^ X{t), fi{t) < k. Let qr{z) be defined by

(5)

= inf + '

N

where the inf is taken over the X satisfying

=

Ш < К Ш ( п ) (* = 1, 2, ...) (1 < i < N ),

N

| ^ < - l | < e ( r ( e ) < w < f c ) ,

i = l

N

\A h{®ъ)\ < £ И *) < Цк), y{k) < k).

г=1

Then qr{z) is a sublinear and homogeneous functional over B(r, 7v);

see [4], p. 116.

Let u(r, k) he the unit vector in É(r, к), i.e. the element 0 for which zn = 1 (r < n < k). We have that qr(u) > 1 so that there exists a linear functional on B(r, k) say F r such that F r(u) = 1 and F r(z) < qr(z) for every 0 in B(r, к),

♦ к oo

ri%) ^г,пАт ^г,п^п

n —r n—1

where ar>n = 0(m < r, n > k) and arn = a'rn (r ^ n ^ k ) . By constructing such a functional for each r we define a matrix A = (аГгП). We have

00

,{6) У а г>п= 1 (r = 1 , 2 ,. .. ) . П — 1

I t is easy to show that for all sequences convergent to zero,

(7) limfirr(0) = 0

and hence

(8 ) limgr(0) = 0 .

From this and (6) it follows that A is regular. Moreover, if A1-lima?1' = 0

\xn\ < Cf(n )(n = 1, 2 , ...) then for sufficiently large г q^x*) < e (the inf in

<5) being less than using X = {xj, xj — 0, j Ф i, xj, x\ j = 1}) so that A (f(n)) => Л* (/(*)).

This concludes our proof.

3. A regular limitation matrix is triangular if am>n = 0, n > m.

D

2. The matrix A ■= (am>n) is an J t matrix if A is tri­

angular and if,

П П* :

I У ат,кЧ ! < K I У an,iksk I

*=I *=1

for some n', n' = n'(n) (0 < n' < n) (n — 1, 2, ...) and for all m (m > n).

The number n ’ depends on n and {sn} but is independent of m.

D

3. A sequence will be called an abbreviated sequence if for some n0, sn — 0 (n > n0).

D

4. A regular matrix, A = (amn), will be called perfect if

to each {sn} limited to zero by A and to every e > 0 corresponds an abbrevi-

ated sequence s'n such that

oo

®PP [ ®'m,n(®n O l < ®*

™ n = 1

In [4], pp. 72-74, it is shown the following theorem.

Th e o r e m

3. A regular Ж matrix is perfect. (See also [2].)

In fact, the proof of this theorem can be used without change to show that if A is an J i matrix and

®UP I , Q'm.rfîn I <~- ® )

*» n = 1

then there exists an abbreviated sequence {s^}, sn = 0 {n < nQ), sn = sn(n

< r0 < n'0) such that

sup I ^ a m>n(sn-s 'n) \ < 5s.

m 71 = 1

Furthermore, since r' and n'0 by the nature of the construction may be chosen as large as we please, let {$"} be a second abbreviated sequence such that n f < r'0. Let t = s' —s ''; then

O O

sup I £ amJ n 1 < 10e and tn = unsn

m 1 n = l

where

un = 0 (n < r'f); 0 < un < 1 <r" < n < л'7);

un = 1 (n f < n < r0) ; 0 < un < 1 (r'0 < n < ri0) ; un = 0 {n > n0) .

With the sequences t(s) and the factors un it is now possible to use the proof in [4], pp. 113-115 and 121, or see [5] to show:

Th e o r e m

4. Two regular J t matrices with 8 ( f( n )) are not g (n)-consistent for any g(n) such that f(n ) = o[g{n)). I f K (s) in (2 ) can be chosen indepen­

dently o f e, they are not f (n)-consistent.

I t is also possible by using the proofs in [4], pp. 121-124, or [5] to show that

Th e o r e m

5. I f A = (am n), В — (bm n) are two regular J l matrices such that A (f(n ))A B [f(n )) spans the bounded sequences, then A and В are not f(n)-consistent.

The only necessary changes in [5] will be e.g. : for

US©

^ K I + |ÿJ > 1/(*»)ЛГ(<0 ( n < n ( r ,e ) ) .

4. As an example for Theorem 5 we first exhibit two matrices which are not J f matrices and then, using the information, we finally illustrate Theorem 5 by two matrices which are J i matrices. Let A = (am>n) and В = (bm>n) be two regular matrices defined by the transformations

-^m . 2 i^2m— 1 ^2m) 1 ? 2 , •••) and

m 2 (^2m ^2m+l) f? •••)>

respectively. It is clear that if s is A limitable to zero, it may be expressed in the form

s = r - f t where t converges to zero and

(9) Let

№г

+

—

(m = 1, 2 , ...).

(10)

— S1 + • • •+

=

(m = 1, 2, . . . ) , (11) T2

— SJ + • • •

II 1 § II rH , 2 , . . . ) . Then,

2f ($ 2 m

-1 + ^2») = 0 (m == 1, 2 ,.. .) , 1

T2m+1) — 0 (m — 1, 2,

$2rn "V 2m ^2m) ^ 2 m + l

1"

= 1, 2, .. .) - If s is bounded, it follows that

A = ( ° ( 4 ?n = (o(n))

and A (n )+ B (n ) includes all bounded sequences. On tl^e other hand, let s2n-i = ni s

n — ~ п ’ч ^ben A-lims = 0 but BAims = \ so that the two matrices are not ^-consistent. Let

= ~ (si + • •. + s m);

m then since

1 2

^2m ( ^ l ••• + A m), C 2m +1 I yC -l-id - ••• + ^ m + "2 s2m+l)>

it is clear that if sn — o{n) and is limited by A it is limited to the same

value by C = (cm>n). I t is also true that C such sequences limited by В to

the same value. Hence A and В are gr(w)-consistent for any g(n) — o(n).

Also

À ( g ( n ) ) + B (g{n))

does not span the bounded sequences as may be seen by a direct examin­

ation of the matrices.

The two matrices of the above example are not J t matrices. However, let P and Q be matrices defined by the transformations

n n n n

p n =

^ P k skl ]? P k ,

(Lkhl

k = l k = l fc = 1 k = l

where p 2n_x = p 2m = 1 /w (m = 1, 2, ...) and qx = 0, q2m = q2m+1 = 1 /m (m = 1, 2 , . . .) .

I t is easy to see that P and $ are both regular J t matrices and P(n)

^A{n ), Q{n) => B ( n ) so that

P ( n ) + Q ( n ) spans the bounded sequences.

The matrices P and Q are n-consistent with A and Б, respectively, so that P and Q are therefore not w-consistent. Clearly,

2 n 2n + l

k*= 1 k = l

and

2 n

\Qzn+l ~^*2ral

j

k = l

The matrices P and Q are again ^(^)-consistent for any g(n) = o(n), for if lsnl = °(n )i

linil^2n+1- P 2n| = 0 .

?г->оо

Thus P and Q give an example of two J t matrices which illustrate the con­

clusions of Theorem 5.

If f(n ) = 1 (n — 1, 2, ...), then (2), (3) and (4) reduce to the defini­

tion of a singularity $ 2, see [3], [4] or [1 ]. On the other hand, if two matrices do not have a singularity $ 2, there is a third matrix G = {cm>n) such that

C(f(n)) => A (f(n ))+ B (f{n ))

for some f(n )/r

and A and В are/(w)-consistent. Hence, if two matrices do not have a singularity S2 they do not have 8 (/(«)) for allf(n ), f(n )/t

The theorems in this paper can also be approached from the point of view of perfect matrices, see [6], p. 40 ff, Sec. 23, 24.

V h ( h - 1)

+ lSli ----К П

References

[1] J . W. B a k e r and G-. M. P e te r s e n , Inclusion of sets of regular summability matrices, Proc. Camb. Phil. Soc. 60 (1964), 705-712.

[2] W. J u r k a t and A. P e y e rim h o ff, Mittelwertsatze bei Matrix- und Integraltrans- formationen, Math. Zeit. 55 (1951), 92-108.

[3] G. G. L o r e n tz and K. Z eller, Über paare von Limitierungsverfahren, ibidem 68 (1958), 428-438.

[4] G. M. P e te r s e n , Regular matrix transformations, McGraw ‘ Hill, 1966.

[5] —, On pairs of summability matrices, Quart. J . Math. (Oxford) 16 (1965), 72-76.

£6] K. Z e lle r and W. B e e k m a n , Theorie der Limitierungverfahren, Second ed., Springer-Verlag, New York-Berlin 1970.