ON THE ALMOST CONVERGENCE OF DOUBLE SEQUENCES. Davor Butković University of Zagreb, Croatia

Transcription

1 GLASNIK MATEMATIČKI Vol. 5(7)(206), ON THE ALMOST CONVERGENCE OF DOUBLE SEQUENCES Davor Butković University of Zagreb, Croatia Abstract. We find necessary and sufficient conditions for transformations of double seuences almost convergent in the sense of G. H. Hardy to double seuences convergent in the sense of F. Pringsheim. The results extend the work of F. Móricz and B. E. Rhoades on transformations of seuences almost convergent in the Pringsheim s sense. The first definitions and investigations of the convergence of double seuences are usually atributted to F. Pringsheim, who studied such seuences and series more than hundred years ago (see [,. 78]). Pringsheim defined what we call the P limit and gaveexamles of convergence(p convergence) of double seuences with and without the usual convergence of rows and columns ([3,. 04 2]). G. H. Hardy ([6]) considered in more details the case of convergence of double seuences where, besides the existence of the P limit, rows and columns converge. F. Móricz discovered an alternative aroach to the Hardy convergence, which significantly influenced the whole theory ([9] []; cf. [2]). Moreover, following G. G. Lorentz ([7]), F. Móricz and B. E. Rhoades found necessary and sufficient (N.S.) conditions under which P almost convergent double seuences are transformed into P convergent double seuences ([2, Theorem,. 285]). In a revious aer [3] the author of this article found conditions under which double seuences almost convergent in the Hardy (H) sense are transformed into P convergent double seuences. The results were not comletely satisfactory because they were obtained under a uniformity condition inherited from the usual H convergence (cf. [3,. 252] and [4,. 4]). 200 Mathematics Subject Classification. 40A05, 40B05, 40C05. Key words and hrases. Almost convergence, double seuences, Hardy convergence, transformation matrices. 75

2 76 D. BUTKOVIĆ In this aer we have N.S. conditions for transformations of double seuences that H almost converge without any further restriction. These conditions are the same as the Móricz-Rhoades conditions for transformations of P almost convergent double seuences. Uniformities of row or column convergence now give various articular cases. Moreover, we have results on transformations of double seuences that H almost converge only by columns or only by rows. For some results that can be reduced to results at [2] and [3] our aroach here gives alternative roofs.. The Hardy convergence Let us denote the set of all double seuences x = ( ), i,j N, of comlex (or real) numbers by s. We consider ( ) as the function on the -coordinate lane: ( ) i N is the j-th row, and ( ) j N is the i-th column of x. Let b be the set of all bounded double seuences from s. A double seuence x from s converges to L if, for every ε > 0 there exists N ε N such that (.) L < ε if i,j N ε. After [,. 78] and [6,. 88] this kind of convergence we call the convergence in the sense of Pringsheim (P convergence). The limit L is denoted by lim and is called the P limit (see [3,. 03]). The set of all convergent double seuences from s is the class c. Bounded and convergent double seuences form the class bc. Starting from double series, both G. H. Hardy and later F. Móricz studied convergent double seuences with convergent rows and convergent columns. Such double seuences besides the P limit L = lim have row limits L j = lim i for every j and column limits L i = lim j for every i. If (L j ) and L exist we say that ( ) converges in the Hardy (H) sense by rows. If (L i ) and L exist we have the H convergence by columns. The class of double seuences that H converges by rows and by columns form the class of H convergent double seuences. It is denoted by rc (regularly convergent, after Hardy [6,. 88] and Hamilton [4,. 30]). The P limit of double seuences from rc is also called the rincial limit. Double seuences from rc are bounded: rc bc ([4,. 33]). Double seuences from rc with eual row and column limits we denote by rcr. For these double seuences L j = L i = L. Subclasses of c of double seuences that converge to 0 are denoted by an n at the end: we have bcn, rcn and rcrn. The last class has all row limits and all column limits as well as the P limit eual to 0. By the definition of rc, (.2) lim = lim i lim j = lim j lim i.

3 ON THE ALMOST CONVERGENCE OF DOUBLE SEQUENCES 77 TheconvergenceofrowstoL j andtheconvergenceofcolumnstol i isuniform with resect to j s, res. to i s (see [4, Theorem 9,. 4] and [4, Theorem 003,. 34]). The convergenceof rows and columns means that for every ε > 0 there exist N ε, N ε N such that for every j (.3) L j < ε if i N ε, and for every i (.4) L i < ε if j N ε. A double seuence x from s is almost convergent (a-convergent) to L if (.5) σ mn = n+ for, converges to L uniformly with resect to m and n. This means that for every ε > 0 there exists N ε N such that, for every m, n N, (.6) σ mn L < ε if, N ε. We denote L by Lim and consider it as the P a-limit of x. The set of all a-convergent double seuences we denote by ac., +m Asingle seuence(x i ) a-convergesif x i convergeswhen uniformly with resect to m N. This notion was introduced by G. G. Lorentz ([7]). It was extended to double seuences i. e. to the class ac by F. Móricz and B. E. Rhoades ([2]). Among others, they roved bc ac b ([2, ]). The double seuence x H a-converges by rows if every row a-converges to L j = Lim i and the P a-limit L of x exists (cf. Lemma.). In such a case L is the rincial a-limit, and L j are row a-limits. Column a-limits and the H a-convergence by columns are similarly defined. The double seuence x H a-converges if it is H a-convergent by rows and columns to the P a-limit. The class of H a-convergent seuences are denoted by rac. The racn is the subclass of rac with the P a-limit 0. The racr is a subclass of rac with eual L, L j and L i for every i,j. If L = L j = L i = 0 we have the class racrn. Lemma. ([2, Theorem,. 32]).Let ( ) P a-converge to Lim = L, and, moreover, let every row ( ) i N a-converge to Lim i = L j. Then (L j ) a-converges and (.7) Lim j Lim i = Lim. Similarly, if the P a-limit and all column a-limits exist, (.8) Lim i Lim j = Lim. Corollary.2 ([2,. 32]). Let a-limits of rows, a-limits of columns and the rincial a-limit of the double seuence x exist. Then (.9) Lim = Lim j Lim i = Lim i Lim j.

4 78 D. BUTKOVIĆ This corollary excludes the existence of the P a-limit if a-limits of L j and of L i exist and are uneual (for P limits cf. [3,. 07]). By row-column uniformities we have a artial inverse of Lemma.. Lemma.3. Let rows of x = ( ) a-converge to L j uniformly, and let the seuence (L j ) a-converge. Then the P a-limit of x exists and x H a-converges by rows. Similarly for uniform a-convergence of columns. Proof. By the uniform a-convergence of rows (.0) L j ε if N ε 2 2 for every j N. By the a-convergence of row a-limits, n+ (.) L j Lim j L j ε if N ε 2. 2 Therefore, (.2) n+ This means that n+ ε 2 + ε 2 = ε. Lim j L j L j + (.3) Lim j Lim i = Lim. n+ L j Lim j L j [ Let A = 2. Transformation conditions ], i,j,k,l N, be a doubly infinite matrix of comlex (or real) numbers. A double seuence x = ( ) s is transformed into a double seuence Ax = y = (y kl ) s by (2.) y kl = j= if the double series (2.) P converges for every k,l N. It means that the artial sums r s j= have the P limit if r,s ([3, (3). 3]). The artial sums for A are called A-means ([2,. 284]). The matrix A is bounded-regular if every bounded and P convergent double seuence ( ) is transformed by the bounded set of A-means to the P convergent double

5 ON THE ALMOST CONVERGENCE OF DOUBLE SEQUENCES 79 seuence (y kl ) with the limit eual to the limit of ( ). The matrix A is strongly regular if every P a-convergent double seuence ( ) is transformed to the P convergent (y kl ) with the limit eual to the a-limit of ( ), and the A-means are also bounded (y P converges boundedly). The above definitions extends to double seuences the notion of regularity and of strong regularity of single seuences (see [7,. 7, 76]). We consider bounded transformations of classes which are between rcrn and b. Necessary conditions that rcrn transforms to b include the existence of C such that (2.2) j= C < + for every k,l N ([4, 5.,. 42]). This imlies that for every bounded y the series (2.) which defines (y kl ) converges absolutely. Because our y are at least bounded, conditions on A always include (with the notation (2.4) below), (v) b. We use the notation (2.3) 0 = akl akl i+,j, 0 = akl akl i,j+, = akl akl i+,j akl i,j+ +akl i+,j+. Series are denoted by i =, j = j=. Also, we give notations for [ ] seuences derived from. The list is a slightly enlarged list ([3,. 254]). The only change is that instead of (ii ) we write (ii ). Also, we have a for. (2.4) (iv) (ii) { ( j) i a ( i) j a { ( j) i a ( i) j a (iv ) (v) i j a (v ) (iii) (i) ( i,j) a { ( j) i (ii ) 0a ( i) j 0a { j i i a j a { ( j) i 0a ( i) j 0a { (iv ) ( j) i 0a ( i) j 0a { i j 0a i j 0a (iv ) { ( j) i a ( i) j a (v ) i j a

6 80 D. BUTKOVIĆ (vi) { j i a i j a (vi ) { j i 0a i j 0a (vi ) { j i a i j a For members of (2.4) which have { and two lines, the first and the second line are denoted by ( ) and ( ) 2. E. g., (vi ) 2 bcn means i j 0 bcn; (vi ) bcn means that (vi ) and (vi ) 2 belong to bcn. Lemma 2.. (cf. [5, (7). 279]). Let (v) b. Then (2.5) 0, 0 tj. j=t it j=t j=t j= i=t j= Proof. The (v) b imlies (v ) b and (2.6) = ( 0 0 i,j+ ) = 0 it. Therefore, (2.7) ( t) Similarly for 0. 0 it j=t. By this Lemma, if (v) b, (2.8) (v ) bcn { (iv ) bcn (iv ) 2 bcn. Lemma 2.2 (cf. [8,. 806, (8) and (3)]). Let (v) b. Then (2.9) 0, 0. j= j= Proof. With a = and C from (2.2), (2.0) (a a i,j+ ) 2 Let (2.) By (2.2) j= j= j= (a a i,j+ ) = D. j= a 2C. a i a i2 + + a it a i,t+ D

7 ON THE ALMOST CONVERGENCE OF DOUBLE SEQUENCES 8 we have (2.3) ( ) ( ) a i a i2 + + a it a i,t+ = a i a i,t+ D. If t, i a i D. Starting with j=s we have i a is D for s = 2,3,... Therefore (2.4) ( j) 0 j= and similarly for 0. By the notation (2.4), if (v) b, (2.5) (vi ) bcn (ii) bcn, (vi ) 2 bcn (ii) 2 bcn (cf. also Remark 2.8). (2.6) Lemma 2.3. Let (v) b. Then and therefore (2.7) ( i,j) 0, j= ( i,j) 0 (iv ) 2 bcn (i) bcn, (iv ) bcn (i) bcn. Proof. By the roof of Lemma 2.2 with instead of i akl for every i we have the first ineuality. The second ineuality is obtained in the same way. The next corollary follows from Lemma 2. and Lemma 2.3. Corollary 2.4. Let (v) b. Then (2.8) (v ) bcn (i) bcn. By Lemma 2.3 (2.9) ( j) 0, ( i) which gives j= j= j= 0,

8 82 D. BUTKOVIĆ Corollary 2.5. Let (v) b. Then (2.20) (v ) bcn (iv) bcn, (v ) 2 bcn (iv) 2 bcn. Remark 2.6. By Corollary 2.5, (v) b and the Móricz-Rhoades (v ) bcn are N.S. for transformations of double seuences from racrn to double P convergent null-seuences. These conditions include (i), (iv) bcn (cf. [2, ]). (2.2) Lemma 2.7. Let (v) b. Then 2 0, Therefore (2.22) j= j= 2 j= j= 0. (v ) bcn (v ) bcn, (v ) 2 bcn (v ) bcn. The results of lemmas are summarized in (2.23). Arrows give imlications if members of(2.23) belong to bcn and ifto everyinitial condition the (v) b is added. E. g. (v ) 2 (vi ) 2 means that i j 0 bcn and j akl b imly i j 0 bcn. i (2.23) Remark 2.8. The schema (2.23) neglects uniformities that in some cases follow from lemmas above. For examle, by (2.20) we have that (v ) bcn imlies (iv) bcn for every j, but also that the convergence by (k,l) is uniform with resect to j. Similarly for estimates by (v), (vi ), (v ), etc.

9 ON THE ALMOST CONVERGENCE OF DOUBLE SEQUENCES 83 Remark 2.9. Instead of (v) b, many aers start with (v ) b (and the notation for our ). Hardy ([6, 4.(2), ]) has a bounded variation for bounded (v ) and (iv ). Mears ([8,. 805]) defines absolutely convergent seuences by (v ) b, with classes ac, arc, etc. Hamilton ([5,. 276]) has bounded i,j= a jointly with other conditions. of (3.) 3. Almost convergence by rows and columns The a-convergenceby rows and columns can be described by σ mn because n+ = σ mj, = σ in. The j-th row a-converges to L j if, for every ε > 0 there exists N ε(j) such that, for N ε (j) and every m, (3.2) σ mj L j ε. The i-th column a-converges to L i that, for N ε (i) and every n, (3.3) σ in L i ε. if, for every ε > 0 there exists N ε(i) such If x H a-converges by rows, L j a-converges to the P a-limit L, and for every ε > 0, there exists N ε such that, for N ε, n+ (3.4) L j L ε. If x H a-converges by columns there exists N ε (3.5) L i L ε. such that, for N ε, Lemma 3.. Let x H a-converges to L. Then, with N ε from (.6), (3.6) N ε,n ε N ε. Proof. Because of (.6), (3.7), N ε n+ L ε. Increasing we obtain (3.4), and increasing we obtain (3.5). As a result, we have (3.6).

10 84 D. BUTKOVIĆ Forx racrn, whichhasatleastone 0, itfollowsthatn ε = N ε = 0 and N ε > 0. If x has eual rows convergent to 0 and at least one 0, we have N ε = N ε > 0, N ε = 0. In what follows we use abbreviations (3.8) (3.9) RC = R = C = mn n= j= mj in n= With L as a P a-limit of x, n+ n+ = = = j= n= mj σmj, inσ. n= y kl L = ( RC ykl) ( R ykl) ( C ykl) For L = 0 as well as for L 0, ( RC L ) + ( R L ) + ( C L ). mn σmn, (3.0) y kl + RC R C = ( RC y kl) ( R y kl) ( C y kl). The subclass of rac with uniformly a-convergent rows and columns is denoted by rac un. This means that (3.2) holds uniformly with resect of j, as well as (3.3) with resect of i. The class racrn un has uniform a- convergent rows and columns to 0. These classes are in [3] denoted by arc, res. by arcrn. Lemma 3.2. Let x be bounded. For every, N, ε > 0 and k,l sufficient conditions on A = [ ] such that (3.) Y kl = ( RC ykl) ( R ykl) ( C ykl) is by absolute value less of ε are (3.2) (v) b; (v ) bcn. Proof. We change the order of summation of the left-hand-side of(3.0). Instead of we write a. If the sum runs over an index that does not aear among indexes, the argument of the sum is constant. For examle, i m=i + a = a. (3.3) y kl = a + j= i= j= i m=i + a

11 ON THE ALMOST CONVERGENCE OF DOUBLE SEQUENCES j= i= j= j n=j + i a j m=i + n=j + a, (3.4) (3.5) R = i a mj + C = j= + + j= i= j= j= i n= j n=j + i i= j= a mj j m=i + n=j + j a in j= i= j= j n=j + i a in i= j= j m=i + n=j + a mj, i m=i + i a mj j a in m=i + n= a in, RC = i (3.6) + + j= j= n= i i= j= j a mn + j n=j + i a mn j m=i + n=j + i= j= a mn. i j a mn m=i + n=

12 86 D. BUTKOVIĆ (3.7) Grouing the corresonding terms of (3.3) (3.6) we get y kl R C + RC = a i j= j + j= [ i= j= + + j= i= j= [ [ n= j= a in + i m=i + j n=j + i a mj i j= (a a mj ) (a a in ) j m=i + n=j + j n= i a mn m=i + n= i j n=j + ] (a a mj a in +a mn ). j ] (a in a mn ) ] (a mj a mn ) Differences of terms of A are finite sums of a µν, 0 a µν, 0 a µν and a µν by (3.8) a mn a in = a mj a = i µ=m i µ=m a mn a in a mj +a = j 0 a µn, a mn a mj = 0 a mν, ν=n j 0 a µj, a in a = 0 a iν, i µ=mν=n j a µν. ν=n Therefore, increasing k and l for every ε > 0 the estimate (3.9) ε y kl R C + is ossible reducing the left-hand-side to the linear combination of (3.20), 0, 0,. j= RC j= By Corollary2.4 and Lemma 2. (v) b, (v ) bcn imly (i), (iv ) bcn. Estimates by using a are given in [3]. For a fixed ν (3.2) i 2 j i m=i + n=j + µ=mν=n j a µν

13 ON THE ALMOST CONVERGENCE OF DOUBLE SEQUENCES 87 is a sum over a triangle with vertices (3.22) (i +,i +), (i,i +), (i,i ). It is dominated by the sum over the rectangle with vertices (3.22) lus (i +,i ): (3.23) i i m=i + µ= + a µν = ( ) i µ= + The same oeration on the lane (ν,n) extends (3.23) to (3.24) ( )( ) i j µ= +ν=j + a µν. a µν. As the result, the corresonding term of (3.7) is dominated by (3.25) ( )( ) su ( )2 ( ) 2 as it is in [3,. 259, (30)]. i j i= j= µ=i + ν=j + su j= a a µν Remark 3.3. The roof of [3, Theorem,. 256] has few obvious misrints. E.g., at. 257 (24) the bracket must be (a mj a ); (25) instead of must be ;. 26 instead of must be / ; 9(33) after akr,lr must stay, the sum 0 must be 2 2nr j=, and instead of = must be. Similarly, at. 263, (40)(d) must start with lim j αkl j ; the words k,l rows at 22 and columns at 24 must be canceled. Instead of [2] 5, [] 6 at. 255 must be [3], [2]. Lemma 3.4. Assume that ( ) a-converges to 0. Then, sufficient conditions for ε for a given ε > 0 as,,k,l are R (3.26) (v) b; (v ) bcn. Similarly, a sufficient condition for C ε as,,k,l in case that ( ) a-converges to 0 is (3.27) (v) b; (v ) 2 bcn. Proof. We start with (3.28) R = RC ( RC R).

14 88 D. BUTKOVIĆ The art of (3.28) in the brackets is (3.29) RC R = (3.30) n= mn ( n+ The kl RC in (3.29) we transform as follows: ( RC = = = + + n= m m mn ( m ( x in + + x i + + x i + + x i + ( a kl ) m + +akl m, n= x i,n+ ) x i, + x i,2 ) x in ). x i ) ( a kl ) m +akl m2 x i2 + x i, ( a kl ) mn + +akl m,n+ x i,n+. Denoting the sum of the first lines above by RC(I), and the last line by (3.3) RC(II) = we obtain (3.32) RC n= ( a kl mn + + ) m,n+ kl kl =Σ RC (I)+Σ RC (II). x i,n+ Slitting the sum with resect toj of R in (3.29)for the art with j and the art with j it follows that (3.33) R (I) = R(II) = j= j= mj mj,.

15 ON THE ALMOST CONVERGENCE OF DOUBLE SEQUENCES 89 Therefore (3.34) with R = R(I)+ = m + R(II) m, + x i, + j= mj (3.35) R(II) = = j= mj m,n+ n= x i,n+. We estimate (3.28) by (3.36) + R RC RC R kl RC + Σ RC (I) + R (I) + RC (II) Σkl R (II). Now we look at members of (3.36). First, we take N ε defined by (.6) and C defined by (2.2). For, N ε we have (3.37) RC = mn n= n+ Cε. For a articular n and ε > 0, by (2.20) and a sufficiently large k,l, (3.38) mn ε for every n (Corollary 2.5 and Remark 2.8). By (3.30), RC (I) is estimated from above by (3.39) ( ) + m m, x i +... x i,.

16 90 D. BUTKOVIĆ By (3.38) and choosing k,l large enough we obtain (3.40) < ε, j =,...,. With (3.4) su = B < +, we have (3.42) mj for every N. Estimating (3.39) we obtain (3.43) B ε, j =,..., RC(I) ( ) 2 B ε = Bε 2. The estimate for R (I) in (3.33) differs from (3.39) having no coefficients,...,. Therefore (3.44) R(I) Bε. To comlete the estimate of (3.36) we estimate the difference of (3.3) and (3.35): (3.45) RC(II) R(II) = ( (a kl mn akl m,n+ ) n= + +( m,n+ m,n+ ) ) x i,n+. For 2 the above differences are sums (3.46) ( mn akl m,n+ )+ +(akl m,n+ 2 akl m,n+ ) ( m,n+ m,n+2)+ +( m,n+ 2 m,n+ ). with, 2,..., terms. For every m in (3.46) there are ( ) 2 terms with 0 mj (counting searately members that in the sum of lines (3.46) result to be eual). The (v ) bcn in (3.26) means that (3.47) lim k,l ( mn m,n+) = 0. n=

17 ON THE ALMOST CONVERGENCE OF DOUBLE SEQUENCES 9 The same euality is valid if we have 0 m,n+ ( = 0,,..., 2) instead of 0 mn. The (3.46) is reduced to the sum of (3.48) ( 0 n= m,n+ x i,n+ ) with = 0,..., 2. The absolute value of (3.48) is less or eual to 0 (3.49) m,n+ B, n= wherebyb = su. Bythe ε Bε -estimatefor(3.47)the(3.49)is ( ). Therefore (3.45) is estimated by (3.50) By (3.50), (3.43) and (3.44) kl RC (II) Σ R (II) Bε 2. (3.5) RC(II) R(II) + RC(I) + R(I) 2Bε. Therefore, with (3.37) we have the estimate for (3.36): (3.52) (C +2B)ε. R This means that Lemma 3.4 holds for kl R. Similarly forσ C. Theorem 3.5. The matrix A = [ ] transforms the sace racrn into the sace bcn if and only if (3.53) (v) b; (v ) bcn. Proof. Sufficiency. With(3.0)and(3.),y kl isestimatedbyestimates for Y kl, RC, R and C. Let, be such that RC < εc, which is ossible by the existence of the P a-limit 0 of σ mn and C from (2.2). Lemma 3.2 gives Y kl as small as we lease with conditions (v) b and (v ) bcn. For R and C, conditions on (v ), res. (v ) 2, are given by Lemma 3.4. Conditions (3.26) and (3.27) assure that, by k,l and every x racrn, both kl R and Σ C converge to 0. The same conditions imly (v ) bcn (Lemma 2.7). Therefore y kl from (3.0) also P converges to 0. Necessity. We assume (v) b and (i) bcn which are N.S. for the transformation of rcrn to bcn. The necessity of (v ) bcn for the sace of all double seuences with columns uniformly a-convergent to 0 and rows nonuniformly a-convergent to 0 is roved in [2, ]. The roof there is given for uniformly a-convergent rows. Moreover, the bounded regularity of

18 92 D. BUTKOVIĆ [ mn ] is assumed. It includes (iv) bcn which is not needed for the necessity roof but follows from (v ) bcn by Corollary 2.5. Remark 3.6. The sufficiency roofin [2] uses (v ) and (v ) 2 together. In our setting they are related to R, res. to C. In the case of the uniformly a-convergentcolumns C is small for large and Y kl is estimated vialemma2.7. Noticethatthe and aredeterminedby(3.37)andhereafter remain fixed. Corollary 3.7. Let subsaces of racrn be sets of all ( ) such that their a-convergence to 0 is (3.54) i. uniform by rows and columns ii. uniform by columns iii. uniform by rows iv. without uniformity restrictions. Conditions on [ ], N.S. for the transformation of the classes above into bcn, corresonding to the resective cases are (3.55) i. (v) b, (v ) bcn ii. (v) b, (v ) bcn iii. (v) b, (v ) 2 bcn iv. (v) b, (v ) bcn. Theorem 3.8. The matrix A = [ ] transforms the sace rac into the sace bc such that lim kl y kl of y = Ax is eual to Lim if and only if (3.56) (iii) bc lim kl = ; (v) b; (v ) bcn. Proof. Sufficiency. We estimate ( (3.57) y kl L = Y kl kl ) ( kl ) ( kl ) Σ RC L + Σ R L + Σ C L. By the Lemma 3.2, (v) b and (v ) bcn are sufficient for Y kl ε if k, l are sufficiently large.

19 ON THE ALMOST CONVERGENCE OF DOUBLE SEQUENCES 93 (3.58) The last three terms of (3.57) are RC L = R L = C L = n= ( + mn n= j= ( + n= mj n= ( + in [ ) mn L, n+ ] ( L) [ ( L i)+ [ n= ) mn L, n+ mn )L. ( ( L j)+ ( n+ )] L i L )] L j L By(iii) bc, lim kl =, increasingk,l wehave n= akl mn L ε 8. With C from (v) b and the P a-convergence of x, the choice of, gives kl 8C and Σ RC L ε 4. By Lemma 3., L i L n+ L j L are also ε 8C. The terms ( L i ) res. ( L j ) σ mn L ε and in R kl L andσ C L a-converge to 0. Lemma 3.4 alies on L i ) as double seuences a-converge to L: x rac and because ( ) and (L i n+ L i = L i converges to L if,. By (3.26) and (3.27) i.e. by (v) b and (v ) bcn, res. (v ) 2 bcn, if k,l, the absolute value of these members becomes ε 8. Therefore, the linear combination of (3.58) is by absolute value ε and y kl L 2ε. Necessity. The N.S. conditions for the a-convergence of seuences which have uniformly a-convergent rows and columns (the class rac un denoted in [3] by arc) are necessary conditions for transformations of seuences from rac. This means that (v) b, (iii) bc with lim kl = and (v ) bcn are necessary. Other conditions, given in [3, Theorem 2,. 36] follow by imlications (2.23). Also, necessary are (v ) bcn because of racrn rac and the Theorem 3.5 above. Similarly to Corollary 3.7, subclasses of rac are characterized by uniformities of row, res. column, a-convergence.

20 94 D. BUTKOVIĆ Corollary 3.9. Let subsaces of rac be sets of all ( ) such that their a-convergence is (3.59) i.uniform by rows and columns ii.uniform by columns iii.uniform by rows iv.without uniformity restrictions. Conditions on A, N.S. that y = Ax belongs to bc with lim kl y kl = Lim are (3.60) (iii) bc, lim kl =, (v) b and, moreover, for cases i iv, (3.6) i.(v ), (vi ) bcn ii.(v ), (vi ) 2 bcn iii.(v ) 2, (vi ) bcn iv.(v ) bcn. Proof. The roof for i is given in [3]. We look at the roof for ii. Sufficiency. For R L in (3.58) we aly (3.60) and (v ) bcn as in the roof of Theorem 3.8. For C L we have (3.62) C L = n= ( + [( n+ in n= mn ) ] L i +(L i L) ) L. Because of the uniformity for ε > 0 by increasing we obtain n+ L i ε for every n,i =,2,... For the transformation of (L i L) we aly (vi ) 2 bcn. Necessity. For C and Y kl transformation conditions follow from [3, Theorem 2,. 26] where N.S. conditions are given for the case i. Conditions (i), (ii) bcn in [3] are suerfluous by Lemmas 2.2 and 2.3. For R, as for Theorems above, necessity follows from the roof for y kl in [2]. Móricz- Rhoades conditions for transformations of the class ac in [2] are conditions of our Theorem 3.8. For the transformations of acn the (iii) b, lim kl = in [2] is suerfluous (cf. also [4, 35. RCN BCN,. 49]). Conditions that follow from [2] for transformations of acn are the same as our conditions for transformations of racrn and therefore also for our racn (Theorem 3.5).

21 ON THE ALMOST CONVERGENCE OF DOUBLE SEQUENCES 95 Proofs of the cases (ii) and (iii) in Corollaries 3.7 and 3.9 remain valid without the a-convergence of rows, res. columns(see Remark 3.6). We denote subsaces of acn and of ac with H a-convergence by columns, res. by rows, by r acn and r ac, res. by r 2 acn and r 2 ac. Therefore, r ac is the class of seuences with the P a-limit L and with columns that a-converge to L i. By Lemma. the L i a-converges to L but row a-limits don t necessarily exist. The class r acr is a subclass of r ac with L i = L for every i and r acn is a subclass of r ac with L = 0. For x r acn and i N the L i are not necessarily 0. The class r acrn is a subclass of r ac with L i = L = 0 for every i. If the a-convergence of columns to L i is uniform, to the designation of the class we add un. If instead of columns we have the a-convergence by rows, instead of r we have r 2. Because the classes r acrn and r ac as well as r 2 acrn and r 2 ac are in the between of racrn and ac, N.S. conditions for transformations of these classes to bcn and bc are conditions of Theorems 3.5 and 3.8. In case that the H a-convergence by columns or by rows is uniform, the corresonding conditions differ from conditions [2]. Theorem 3.0. The matrix [ ] transforms the sace r acrn un into the sace bcn if and only if (3.63) (v) b, (v ) bcn. Conditions N.S. for the transformation of r 2 acrn un into bcn are (3.64) (v) b, (v ) 2 bcn. The matrix A = [ ] transforms r ac un into bc such that lim kl y kl of y = Ax is eual to Lim if and only if (3.65) (iii) bc, lim kl =, (v) b, (v ) bcn, (vi ) 2 bcn. The N.S. conditions for the analogous transformation of r 2 ac un are (3.66) (iii) bc, lim kl =, (v) b, (v ) 2 bcn, (vi ) bcn. The N.S. conditions on A for the transformation of r acr un, res. r 2 acr un, into bc with lim kl y kl = Lim are (3.63) res. (3.64), with added (iii) bc, lim kl =. The N.S. conditions on A for the transformation of r acn un, res. r 2 acn un, into bcn are (3.65), res. (3.66) without (iii) bc, lim kl =. Proof. With inclusions commented in the introduction of classes with r and r 2 the various cases above are deduced by the corresonding cases in Corollaries 3.7 and 3.9. See also Corollary in [3,. 26]. Remark 3.. In accordance with the notation for classes in Theorem 3.0we can denote classesin Corollaries3.7(ii), (iii) and 3.9(ii), (iii) by racrn un, racrn un 2 and rac un, rac un 2. The necessity counterexamle in [2] is from racrn un 2.

22 96 D. BUTKOVIĆ Acknowledgement. The author wish to thank the referee for his careful reading of the manuscrit, correcting many errors and ointing on incomlete roofs. References [] T. J. l A. Bromwich, An introduction to the theory of infinite series, ed. 2, Cambridge 926 (rer. Macmillan, London, 955). [2] D. Butković, On regular almost convergence, Math. Commun. (996), [3] D. Butković, On almost regular convergence, Acta Math. Hungar. 82 (999), [4] H. J. Hamilton, Transformations of multile seuences, Duke Math. J. 2 (936), [5] H. J. Hamilton, On transformations of double series, Bull. Amer. Math. Soc. 42(936), [6] G. H. Hardy, On the convergence of certain multile series, Proc. Cambridge Phil. Soc. 9 (97), [7] G. G. Lorentz, A contribution to the theory of divergent seuences, Acta Math. 80 (948), [8] F. M. Mears, Transformations of double seuences, Amer. J. Math. 70(948), [9] F. Móricz, On the convergence in a restricted sense of multile series, Anal. Math. 5 (979), [0] F. Móricz, Some remarks on the notion of regular convergence of multile series, Acta Math. Hungar. 4 (983), [] F. Móricz, Extensions of the saces c and c 0 from single to double seuences, Acta Math. Hungar. 57 (99), [2] F. Móricz and B. E. Rhoades, Almost convergence of double seuences and strong regularity of summability matrices, Math. Proc. Cambridge Philos. Soc. 04 (988), [3] A. Pringsheim, Elementare Theorie der unendlichen Doelreihen, Sitzung-Berichte der math.-hys. Classe der Akad. der Wissenschafften zu München 27 (897), [4] V. G. Qelidze (V. G. Chelidze), Nekotorye metody summirovani dvo nyh r dov i dvo nyh integralov (Russian), MR g:4000: Summability methods for double series and double integrals, Izdat. Tbilis. Univ., Tbilisi, 977. D. Butković University of Zagreb Faculty of Electrical Engineering and Comuting Unska 3, 0000 Zagreb Croatia dbutkovic@gmail.com Received: Revised: &