MATRIX TRANSFORMATIONS OF SUMMABILITY AND ABSOLUTE SUMMABILITY FIELDS

DISSERTATIONES MATHEMATICAE UNIVERSITATIS TARTUENSIS

i^%

MATRIX TRANSFORMATIONS OF SUMMABILITY AND ABSOLUTE SUMMABILITY FIELDS

OF MATRIX METHODS

by

ANTS AASMA

4

TARTU 1993

DISSERTATIONES MATHEMATICAE UNIVERSITATIS TARTUENSIS

%ПШ#

MATRIX TRANSFORMATIONS OF SUMMABILITY AND ABSOLUTE SUMMABILITY FIELDS OF MATRIX METHODS

ABSTRACT OF THE INVESTIGATIONS PRESENTED TO OBTAIN THE ACADEMIC DEGREE OF A DOCTOR OF MATHEMATICS

7

by A. Aas ma

TARTU 1993

T h e r e s u l t s o f t h e i n v e s t i g a t i o n s p r e s e n t e d i n t h e p a p e r w e r e o b t a i n e d i n T a r t u U n i v e r s i t y ( 1 9 8 3 - 1 9 8 6 ) a n d i n t h e P e d a­

g o g i c a l U n i v e r s i t y o f T a l l i n n ( 1 9 8 6 - 1 9 9 2 )

S c i e n t i f i c c o n s u l t a n t

C a n d . P h y s . a n d M a t h . , A s s i s t . P r o f . M . A b e l ( T a r t u ) . O f f i c i a l r e f e r e e s :

C a n d . P h y s . a n d M a t h . , P r o f . T . L e i g e r ( T a r t u ) ,

C a n d . P h y s . a n d M a t h . , A s s i s t . P r o f . A . T a l i ( T a l l i n n ) , C a n d . P h y s . a n d M a t h . , P r o f . S . B a r o n ( B a r - I l a n U n i v e r ­ s i t y , I s r a e l ) .

T h e d i s s e r t a t i o n w i l l b e d e f e n d e d o n M a r c h 2 5 , 1 9 9 3 a t 1 6 . 0 0 a t t h e C o u n c i l H a l l o f T a r t u U n i v e r s i t y (Ü l i k o o l i 1 8 , T a r t u E E 2 4 0 0 , R e p u b l i c E s t o n i a ) .

T h e m a t e r i a l p r e s e n t e d t o g e t t h e a c a d e m i c d e g r e e o f a D o c t o r o f M a t h e m a t i c s i s a v a i l a b l e a t t h e T a r t u U n i v e r s i t y L i b r a r y . T h e a b s t r a c t h a s b e e n m a i l e d t o l i b r a r i e s o n F e b r u a r y 2 6 , 1 9 3 3 .

S e c r e t a r y o f t h e C o u n c i l

D r . P h y s . a n d M a t h . , P r o f e s s o r

JU

JL

C

M

A n t s A a s m a w a s b o r n i n 1 9 5 7 . G r a d u a t e d f r o m t h e T a r t u U n i v e r s i t y i n 1 9 8 0 , p o s t g r a d u a t e s t u d e n t i n 1 9 8 3 - 1 9 8 6 i n t h e s a m e u n i v e r s i t y , a t e a c h e r i n 1 9 8 7 - 1 9 9 2 a n d s i n c e 1 9 9 2 a l e c t u r e r i n t h e P e d a g o g i c a l U n i v e r s i t y o f T a l l i n n . A u t h o r o f 8 s c i e n t i f i c p a p e r f r o m w h i f a h 7 a b o u t t h e t h e o r y o f s u m m a b i l i t y .

© A n t e A a s m a , 1 9 9 3

INTRODUCTION

The considerable dissertation belongs to the theory of summability of series and sequences . The aim of this work is to study matrix transformations from the summability Cor absolute summabilityD field of a matrix method of summability into the summability Cor absolute summabilityD field of another matrix method of summability. In the case when the transformation matrix has a diagonal form the considerable problem is reduced to the problem of summability Cor absolute summability:) factors, which have been widely investigated Cef. , for example, the S. Baron's monography^v[36] and the articles С7,8,9,10-14,17-88,34,35,371Э.

Up to now for solving the above-mentioned problem both functional analitic methods and the methods which use mainly the results from the classical theory of summability have been considered by different authors. In [15,16] the necessary and sufficient conditions for the matrix that it would transform a sequence space into another sequence space have been obtained.

These conditions have been given by the properties of certain kind dual Сso called the y-dual and the second y-dual) spaces of these sequence spaces. In the case when these sequence spaces are summability fields of summability methods the results of [15,163 are available but to describe the dual space and the second dual space for given summability field of summability method is complicated enough , not to speak about its properties.

Therefore, for solving the above-mentioned problem methods which use only the properties of considerable methods of summability and the properties of continuous linear functionals on summability fields of these methods of summability are considered.

The first result in the case of non-diagonal matrix С malniy by the classical methods of theory of summability) obtained L.

Alpâr In 1978 Cef. [41Э. He found the necessary and sufficient conditions for matrix M - C"*nk) that the transformation

3

- £ mnkXk C1)

transforms each convergent series to C-summable series Ц^УП

for « > О. After that, in 1Э80, he found the necessary and sufficient conditions in order that the matrix M transforms each C^-summable series to C^^-summable series for ot. ft Ž О Cef. [53) and, in 1982, he generalized the above-mentioned result looking now at the method Cp instead of the method C^a ^ Cef. С63). In addition, in 1986, B. Thorpe generalized the result of L. AlpAr С also mainly by the classical methods of the theory of summability) considering now instead of the method an arbitrary normal method of summability В Cef.[29]). In this paper he found also the necessary and sufficient conditions in order that the matrix M would transform each C^a-summable series to B-summable series in the case when -1< ot < О and В is a normal method.

In the present dissertation this problem is considered more generally studing matrix transformations from a summability Cor an absolute summability) field of a method A into the summability Cor an absolute summability) field of another method В in the case when Л is a regular perfect or a reversible method and В is an arbitrary triangular or an arbitrary method. The cases when A or A and В both are Cesàro or Riesz methods as applications are considered.

All results of dissertation have been obtained in the period of 1984-1990 and Introduced in the seminars of the department of mathematical analysis of Tartu University Cin 1984-1986, 1989 and 1991), at the conferences "Problems of pure and applied mathematics" С1985, 1990) and "Methods of algebra and analysis"

C1988) and in the seminar of theory of function in Ural State University С1988).

The main results of the present dissertation have been published in [1-3.31-33].

4

MAIN RESULTS

The present dissertation includes an introduction. two chapters, which both consist of three paragraphs, references and the table of basic symbols. All this material has been presented on 78 pages.

In the Introduction a short review of the subject, purposes and the structure of dissertation are given .

In the first chapter basic notations and notions are given which often are used later on. As follows we shall give a short account of it.

Let M * C^mnk> be a matrix over C. We shall often write (1) in the form y = rtx or y = (W^x) where M^x = y^ as usual.

Moreover, let о» be the set of all number sequences, in which the algebraic operations have been defined coordinate-wise, * and fi be the subsets of u and A

following notations:

(°<nk5 be a matrix over C. We use the

bv

bi»°

XA

CA

bM and

C*. aO

X - ( xn>

X -C xn>

x - ( xn)

X - ( xn)

x - ( xn>

X = <xn)

X - ( xn) x ж ( xv)

х е ш a n d t h e r e e x i s t s f i n i t e l i m i t l i m x > ,

n n

x e с and lim x^ - О >,

х е ш a n d t h e s e r i e s E ^ x ^ i s c o n v e r g e n t } , x e о, E |x -x I < OO and x =0},

n n—1 -*

x « bv and lim x^ * О }, x e u and (Л^х) e * }, (Л^х) « ш and lim A^x =» 0>,

<v> e " and ^k=i' nk к C(l) M ^k) I e С and Mx e yu for each x e * >.

The spaces с^ and bv^ are usually called a summability field and an absolute summability field of method A respectively.

Definition.

methods A

Let M С^лпк ) be a matrix. W« say that two I <

lim A x _{n n} • lim TL ft .M, _{n n k к}

( < x ) a n d В * Cßn k) a r e M - c o n s i s t e n t i f

for each x e с

S

2

It, is easy to see what M—consi stency of methods of summability coincides with the ordinary consistency of them if M is an identity matrix.

The main results of the dissertation are proved in §2 and §3 of Chapter I The necessary and sufficient conditions for the Matrix transformations from с ^ into are obtained by the method of Peyerimhoff Cef., for example, 1243 3, but for the matrix transformations from с A into cß ,from с ^ into buß .from bv^ into e, and from bv^ into - by the inverse transformation method Cef. , for example, [34]3.

Now we shall give the results of Chapter I in greater detail. For that let e = (1,1,...) and ex = Ç0 0,1,0,...) where 1 is in k-th position. In §2 it is assumed that A = is a series-to-sequence and = (a ) is a sequence-to-sequence transformation. If Д = •(e0,ei,... }• and AU-{e} are fundamental sets for с^ and respectively then the methods A and It are called perfect. Here it is assumed that Cor equivalently with it Cy Э and с ^л are BK-spaces Ci.e. the Banach spaces where coordinate-wise convergence hoidsZ>. The topologies in с ^ and in Cy are defined respectively by the norm IxB^ = sup 1/4^x1 С for each x € с Э and by the norm 8x11 о = sup|%l x| Сfor each x e cfb. Л С П n «4 It is assumed that В = ( is a triangular matrix over С , H =

* is an arbitrary matrix over С and G = (g is the product of above-mentioned matrices, i.e.

gnk ⁼ С Р^пштак •

• =o

Using the method of Peyerimhoff С which is based on the properties of continuous linear functional s on and с the matrix transformations from into cg and from сA into are studied in §2 in the case when and A are the regular perfect methods.

Theorem 1. Let V = be such a regular perfect method that с у is a В K-s pace, В = ( ) be a triangular method and H =

= be an arbitrary matrix. Then M € (cy,c^ß) if and only if

« cs for each s с IN ,

1) there exist finite limits lim g^ = g ,

2) there exists finite lim.it lim = g and there exist such functionals f ( e (c^)' that

Г m i f к < I .

/

C*> -

l. о . if h > I .

"

°.

- «4

where the functionals F haue been defined on by Fn(x°) = £ /3^no/^e(x°)

/e(x°) = /el(x°) •

Moreover, if in addition g, = О arvi jç = 1 , then, the methods 41 and 5 are M-consistent.

An analog of Theorem 1 for a regular perfect method A too Is presented In §2 .

Let now be such a regular method for which cf, is a BK-AK-space. It means that c^ о is simultaneously a BK-spa.ce and an AK-space Ci.e. Ac cf*. and lim Их""0 - xli =0 for each x = fx, ) s

41 n к

€ where xCnl = ( ^x0 » • . . , x^, О,. . . ) or С cf. [ ЗО] , p., 1763 in Су the weak convergence by the sections is valid , i.e.

lim I/(xM) - /(x) I = О for each x = (xfc) <= and / e (c^)»

where (c^)' is a topological dual space of c^3. It is easy to see that a regular method is perfect when is a BK-AK-space, but for each regular perfect method It the space c^ is not necessarily an ЛК-space Cef. , for example, С 30], p. 214 - 2153.

Theorem 2. Let У = (a^) be such a regular method that c^ is a BK-AK-space, В = (/J k) be a normal method and И = (m. fc) be an arbitrary matrix. Then M e ( c^e,, U с ^ny ) if and only if conditions 1) ⁰ arui 2) о/ Theorem. 1 hold and there exist functionals F^ € (c^)' such that

^ Ce")

2*

ВГ 8. о = 0(1).

п (Су)

Moreover, if, in addition, = О and g = 1 , then the methods

%t and В are M-consis tent.

Using the general form of continuous linear functional on e° and on с^ for reversible methods It and A Ci.e. for such methods К and A for which the systems of equations 2 = IZx and я = Ax have a unique solution for each г e cD it is easy to find conditions that M € (c^.c^) and W e (c^,c^ß) in the case when У and Л are regular perfect reversible methods.

Corollary 1. Let 4t = (a ) be such a regular rex>ersible method that is an AK-space, В = ( ß ^ ) be a normal method and M = C"ink) be an arbitrary matrix. Then M e (c^.c^) i/ and only i/

the condi tions 1) and 2) of Theorem 1 have been fulfilled and there exist series with the property Ц. '^T I ⁼ such that

enk = E Tnrark •

Corollary 2. Let У = C^cink) be a regular reversible perfect method, В = ( /3^^к ) be a triangular method and M = (m. ^ ) be an arbitrary matrix. Then M e (c^.c^) i/ and only if conditions 1 ) and 2) of Theorem 1 and condi tions

3) there exist series T r" r I r with the property T !т"r 1 ч I Г ! = © ( а 1 ) such that

Г m , if к < I , E,<,^a,k =

\ '

4) Г j D _r ! = ©(1 ) where numbers D have been defined bv

1 nr ¹ nr *

Dnr ⁼ С /?^пвт"

л.к = ErT>rk

(here the existence and absolute convergence of series r*

have been guaranteed by condition 3)J>

have been fulfilled.

8

The matrix transformations from с^ into с^g , from с^ into bUg , from bt>^ into с g and from bu^ into bvß are studied by inverse transformation method in §3 of chapter I in the case when A is a reversible method and В is a triangular method or an arbitrary method. To describe these matrix transformations for the case of triangular method В the necessary and sufficient Сbut for the case of an arbitrary method В only the sufficient) conditions are found. It is well-known that с^ is a BK-space if A is a reversible method. Therefore in this case the members of each sequence x = (x^fc) e с^ are continuous linear functionals on с ^A . Thus the member s of each sequence x = (x^fc) e с^ may be represented in the form

where = Л^х . /и = lim and the sequences (Юкп) С for fixed rO and ( ) are the solutions of the system г = Ax for г ^ = <5^ and

= <5^u respectively С here 6^ = 1 for I - n and ^^lr>= ® for l x x rO.

We shall consider the case when В is a triangular method. Then

for each x e с ^ where у = Cv^) = (^^kx)- By (2), (3) and some well-known results from the theory of summability Сfor example, the theorems of Koi Ji ma-Schur, Hahn and Knopp-Lor entzD the necessary and sufficient conditions for the above-mentioned four types of matrix transformations are proved. Here we present some of them.

Theorem 3. Lei A =( b e a reversible method, В = ( ft ) be a tri angxi I or me t hod and M = Ç ) be an orb i t rary ma (rix. Then M e (c^,cß) i/ and only i/

1) there exist finite Limits lim = W k , 2) series are convergent,

( 2 )

Let.

(3)

3> SJO - »„CD •

4) there exists finite Limit 1 im 2^8 kT7k = S .

9

S) there exist /int te limits lim

=) ïblr^i - OC1) where

P

= £ m. 7) .

pk ito nl lk

Moreover, t/ s О and # = 1 then the methods A and В are M-consistent.

Theorem 4. let Л = C«nc) be a reversible method , В = (/9^.) be a triangular rnethod and M = (r>v ) be an arbitrary matrix. Then W e <bv^,bi>e) if and only if

1) there exist finite limits 1 im /Y^^k = , 2) series ^are convergent,

3) E = on(D . k = o ^pk

4) C»k) e bvG

=> - w l * I

Some analogs of Theorems 3 and 4 for transformations from сA

into trug and from bv^ into c^ß are also proved.

At the end of this chapter we consider the case when the method В is arbitrary. Let G = (#^nk) where

S . - 71 ft m .-_{Tik т na ak}

Then (3) is not necessarily valid for each x e с „ or x e bv where y = (W^fcx) . In the present dissertation the necessary and sufficient conditions for it are found. Using these conditions we have

Theorem 5. Let A = (a^) be a reversible method , В = ( /9^ ) be an arbitrary method and M = (^mnk5 be an arbitrary matrix.

Moreo-oer, let Ekl/5nkl 00 » mnk = ®k(l )• exist finite limits lim = >f^nk ,

E Vk ^* k=0

10

and one of the conditions

• °сч

or t

holds . Then there exist finite limits lim ys sk ⁿ = у , . nk Here M e e (c 4> cg) i/ condi t ions 4) - 6) of Theorem 3 and M e (bu^„bi>5) i/

conditions 4) arid 5) of Theorem 4 have been fulfilled.

The same kind of analogs of Theorem 5 hold for the classes (c^,buß) and (bu^, buß) too.

APPLICATIONS

Now we shall consider the cases when A or A and В both are Cesàro C§1 of Chapter IID or Riesz С§2 and §3 of Chapter 115 methods. Let A01 n - |[n J ^n+0<] for each <x e С and n e IN. We keep in mind that the series-to-sequence method of Cesàro of order ot Cet e e Cn<-1 I -2,. . . }0 , shortly c" method, is defined by the matrix (°<rlc) where

(Лы

' { =

and the sequence-to-sequence method of CesAro of order ot, shortly ОТ* method, is defined by the matrix (a^) where

* if h < n . if h > n .

For the description of the main results of §1 of Chapter II we put

nk

Г А°"

• l o -

=

^

Г**

'

for each bounded number sequence (^ek) and for each <x e C. If Re« > -1 or ot ж —1 then Ц I < 0 0 • ®У Corollary 2 w»

have

11

Theorem в. Let В = (ßnk) be a triangular method . M - C"»nk5 be an arbitrary matrix and. a be such a complex number for which Reot > О or ot = 0. Then И s (c , c^ß) i/ and only i/

ctr"

_ , -Reot.

= °nCM 5.

E (к + I < OO. (4)

к I ¹

E (k + 1)^1^j = ©(l)

and conditions 1) and 3) of Theorem 1 haue been fulfilled .

In addition, by Theorem 3 we have

Theorem 7. lei В = ( ßnlr) £>e a triangular method, M = (яг fc) be an arbitrary matrix and о/ be such a complex number for which Reex > О or <x • О. Then M e (e ,с-) i/ and only i/

С* в

nk = °ПС *"***>•

Е с* + <- OO. (5)

_ , . . „ .Reoi I .a» i I , . E С* + i ) |\ j = ОО ) and condition 1) of Theorem 1 has been fulfilled.

Moreover , if g^ = 1 then the methods c" and В are M-consistent.

For a normal method В the condition (4) in Theorem 6 and the condition (5) in Theorem 7 are redundant. Some generalizations of the results of С 4-6,293 follow from Theorem 7 in particular .

Furthermore , let (p^) be a sequence of non-zero complex numbers, P = p + ...* p _{л О л} * О for each n e IN, P = O, CR,p ) =

—1 л

= C«^nk) and (9t, = Cank> be respectively the series-to-sequence and sequence-to-sequence Riesz methods generated by (Pn)» i.e.

-/P_ i/ к < n , if к > n

Г

- \v

" 1°

12

and

if h < n , if h > n .

•С'

Мог eover, 1 et

Д a , = « , - « ,

nk nk n.k-f-1

and В = (^^nk) be an arbitrary triangular method. By Corollary 1 and Theorems 3 and 4 hold

Theorem 8. Let ( 9t, p_ ) be a regular method , В = ( /3_^ь) be a I me t hoc?

<9t.p_ycB:

nk' normal method and M = (m be an arbitrary matrix. Then M e e (c._ ^ ,c ) i/ and only i/

'nk = » (Pv>-

and conditions 1) and 2) of Theorem 1 have been fulfilled.

Theorem 9. let (R, p_) be a conservât ixje method, В = 'nV

nk be a triangular method and M = (m ) be an arbi trary matrix.

Then M e y^cB~> ^£/ anjd only if i Amnk,

' 0 0 • <=>

• W'

-О С¹' and condition 1) of Theorem 1 has been fulfilled.

Moreover, if = 1 then the methods (R,p^) and В are M—consistent.

Theorem 10. £et be an absolutely conservatix>e method, В = С/?лк> be a triangular method and M = (m. ^) be an arbi trary matrix. Then M e (bv^{z D} - , bu_) i/ and only if

v. ^K-Pⁿ J о

Pk^mnk ⁼ °n( pk>'

Pk^nk = C7)

13

Ц>пк * and

- «„-.л" - °«v where ^ k =0.

It is shown that for a normal method В the condition (6) in Theorem 9 and the condition (7) in Theorem 10 are redundant.

Moreover, for the case of the triangular method В the necessary and sufficient conditions for transformations from с . _{( R . P}^D . into

n) bv„ and from bv,- x into cB are found too .

в С *Pn' By Theorem 5 we have

Theorem 11. Let (R, p ) be a conservative method. , В = ( ) be a method which satisfies the condition. E^lel^^nlel ^{< 00}• and A# =

= (m. ) be an arbitrary matrix. If

= o<p,).

Цс'^рк^А—rl

П | k pk '

'Ч

P b

I P. л—TT ! = о<1) к ¹

and condition 1) of Theorem. 1 has been fulfilled then M

Theorem 12. Z.et ( R. p ) be an absolutely conservative method, В = ( p } be a method which satisfies the condition I ß ^k I < OO and M = (m. be an arbi trary matrix. If

= ®C"k>-

Pk^nk = - »n-t.kl = «V

W**nk - *„-4.k>" -^k>

where # 4 k = О then И e (bv^ R ^ y bvfî).

1 4

The sufficient. conditions for the transformations from c,„ . into bvn and from bv. _ x into cD are also given . The

(R,pn) В (RPn) В

case when В is a Riesz method is considered separately.

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MAATRIKSMÖETLUSTE SUMMEZRUVUS VA LJ ADE JA ABSOLUUTSE SUMMEERUVUSE VÄLJADE MAATRIKSTOSEhDUSED

Ants Aasma

RESÜMEE

Antud väitekirjas vaadeldav probleem kuulub summeeruvus- teooria valdkonda. Olgu A = (°<nk) Ja В = C/?^nk) maatriksmenetlused üle С ning M ^ш С^жпк) maatriks üle С. Peale selle, olgu с^ Ja bv^

vastavalt menetluse A summeeruvusväli ning absoluutse summeeru- vuse väli. Lisaks eeldatakse, et on BK-ruum. Väitekirjas uuritakse maatriksteisendusi ruumidest vöi bvruumidesse c^ß vöi bv^ß . Peyerimhoff! meetodiga leitakse tarvilikud Ja piisavad tingimused selleks, et maatriks M teisendaks ruumi ruumi cg

Juhul, kui A on regulaarne perfektne menetlus Ja В on kolmnurkne menetlus. Seejuures Jada-Jada teisendusega antud menetluse А Jaoks vaadeldakse eraldi Juhtu, kus c° Cmenetlusega A nulliks summeeruvate Jadade ruunO on ЛК-ruum Ja В on normaalne menetlus.

Pöördteisenduse meetodiga leitakse aga tarvilikud Ja piisa­

vad tingimused selleks, et maatriks M teisendaks ruumid vCi bvA ruumidesse c^ß vöi bvg Juhul, kui A on reversiivne menetlus Ja В on kolmnurkne menetlus. Suvalise menetluse В korral leitakse nimetatud nelja tüüpi teisenduste Jaoks ainult piisavad tingimused.

Rakendustena vaadeldakse Juhtumeid, kus menetlus A vöi menetlused A Ja В mölemad on kas Rieszi kaalutud keskmiste menetlused vöi Cesàro menetlused.

TÜ 93.52.120.0,89.1,0.