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Core theorems for subsequences of double complex sequences.

1 introduction

In [9], Miller and Patterson proved a theorem, that answers the question mentioned in the abstract, for double sequences whose entries are real numbers. In the introduction of that paper results for single sequences of reals are presented as motivation (see [9])). In this paper we answer the question asked for double complex sequences mentioned in our abstract. All double sequences, in the remainder of this paper, will be assumed to have complex entries.

2 Definitions and Preliminaries

Definition 2.1. A double sequence w = ([w.sub.u,v]) has Pringsheim limit L (denoted by P - Urn w = L) provided that given any [epsilon] > 0 there exists N [member of] N such that [absolute value of [w.sub.u,v] - L] < [epsilon] whenever u,v > N. We shall describe such a w more briefly as "P-convergent".

Definition 2.2. A double sequence w is called definitely divergent, if for every G > 0 there exist two natural numbers [n.sub.1] and [n.sub.2] such that [absolute value of [w.sub.u,v]] > G for u [greater than or equal to] [n.sub.1], v [greater than or equal to] [n.sub.2].

Definition 2.3. The double sequence z is called a double subsequence of the sequence w provided that there exist two strictly increasing index sequences ([n.sub.f]) and ([k.sub.f]) such that [z.sub.j] = [w.sub.nj,kj], and z is formed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is: [z.sub.1,1] = [z.sub.1],[z.sub.1,2] = [z.sub.2],[z.sub.1,3] = [z.sub.5],[z.sub.2,1] = [z.sub.4],[z.sub.2,2] = [z.sub.3] etc.

Definition 2.4. A number [beta] is called a Pringsheim limit point of the double sequence w = ([w.sub.u,v]) provided that there exists a subsequence z = ([z.sub.u,v]) of w = ([w.sub.u,v]) that has Pringsheim limit [beta], i.e. P - lim z = [beta].

The double sequence w is bounded if there exists a positive number M such that [absolute value of [w.sub.u,v]] < M for all u and v. A two dimensional matrix transformation is said to be regular if it maps every convergent sequence onto a convergent sequence with the same limit. The Silverman-Toeplitz theorem [15], [16], characterizes the regularity of two dimensional matrix transformations.

Let A = ([a.sub.m.n.u.v]) denote a four dimensional real matrix. We obtain a summability method that maps the double complex sequence w into the double complex sequence Aw where the mnth term of Aw is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in the sense of Pringsheim convergence, for all m, n = 1,2,3.... In this case we call the terms [(Aw).sub.m,n] the A-means of w. For a reference on the above and what follows see Moritz and Rhoades [13] . Moreover, we say that a sequence w is A-summable to the limit t if the A-means exist for all m,n = 1,2,3 ..., and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

in the sense of Pringsheim convergence.

Definition 2.5. The four dimensional real matrix A is said to be bounded regular if every bounded P-convergent double complex sequence w with Pringsheim limit t, is also A-summable to t and the A-means of w are bounded.

Theorem 2.6. ([6],[13],[14]) Necessary and sufficient conditions for A to be bounded regular are:

(1) : [lim.sub.m,n][a.sub.m,n,v] = 0 for each u and v;

(2) : [lim.sub.m,n] [[summation].sup.[infinity],[infinity].sub.u,v=1,1][a.sub.m,n,u,v] = 1;

(3) : [lim.sub.m,n] [[summation].sup.[infinity].sub.u=1][absolute value of [a.sub.m,n,u,v]] = 0 for each v;

(4) : [lim.sub.m,n] [[summation].sup.[infinity].sub.v=1][absolute value of [a.sub.m,n,u,v]] = 0 for each u;

(5) : [[summation].sup.[infinity],[infinity].sub.u,v=1,1][absolute value of [a.sub.m,n,u,v]] is P-convergent; and

(6) : there exist positive integers A and B such that [[summation].sub.u,v,>B] [absolute value of [a.sub.m,n,u,v]] < A for each m, n.

We now define the concept of a [lambda]-rearrangement for double sequences (from [9]).

Definition 2.7. A mapping [phi] : NxN [right arrow] NxN is called a [lambda]-rearrangement, [lambda] > 1, of NxN if it is a one to one, onto function such that [phi](u,v) = (u,v) for (u,v) [member of] NxN ([lambda] - wedge) where the ([lambda] - wedge) = (u,v) : 1/[lambda] [less than or equal to] u/v [less than or equal to] [lambda]}. A [lambda]-rearrangement of w, a double sequence is a double sequence of the form ([w.sub.[phi]](u,v))u,v] where [phi] is a [lambda]-rearrangement of NxN.

Definition 2.8. The double sequence z is called a rearrangement of the double sequence w provided that there is a one-to-one, onto [phi] : NxN [right arrow] NxN such that for each (u,v), [z.sub.(u,v)] = [w.sub.[phi](u,v)].

In [8] the following is proved.

Theorem 2.9. If w is a bounded sequence and A = ([a.sub.m,n]) regular matrix summability method satisfying [lim.sub.m]([sup.sub.n][absolute value of [a.sub.m,n]]) = 0, then there exists a subsequence z of w such that each t in the core of w is a limit point of (Az). Here the core of w equals [lim inf w, lim sup w].

Definition 2.10. A = ([a.sub.m,n,u,v]) is said to satisfy condition (S) if the double sequence ([sup.sub.u,v][absolute value of [a.sub.m,n,u,v]])m,n is Pringsheim convergent to zero.

Definition 2.11. If w is a double sequence, then we use the following notation:

(a) : C(w) denotes the smallest convex set containing all limit points of w. C(w) is called the core of w;

(b) : L(w) denotes the set of all limit points of w;

(c) : D(w) denotes the set of all complex numbers of the form [[summation].sup.n.sub.i=1][[alpha].sub.i][t.sub.i];[[alpha].sub.i] [greater than or equal to] 0 for all i, [[summation].sup.n.sub.i=1][[alpha].sub.i] = 1, and [t.sub.i] [member of] L(w) for all i.

Remark 2.12. It is easy to show that C(w) = D(w) and it is a closed set.

3 Results

We will prove analogues of Theorems 3.1 and 3.2 from [9], for double sequences with complex entries.

Theorem 3.1. If A = ([a.sub.m,n,u,v]) is a four dimensional bounded regular real summability method satisfying condition (S) and w = ([w.sub.u,v]) is a bounded double complex sequence, then there exists a double subsequence z of w such that each t [member of] C(w) is a Pringsheim limit point of Az.

Proof: If w is Pringsheim convergent then the result is immediate by the regularity of A. If C(w) is a line segment in the complex plane then a minor modification of the proof when w = ([w.sub.u,v]) is a bounded real double sequence, found in [9], is applicable. Therefore we will only consider the case when C(w) contains three non-linear points in the complex plane.

We note again that it is an easy exercise to show that C(w) = D(w). Now, if C(w) contains three non-linear points, it is easy to see that there exists a complex sequence ([s.sub.n]) that is dense in C(w) and such that each [s.sub.n] is an interior point of C(w). Let ([t.sub.n]) denote the sequence [s.sub.1],[s.sub.1],[s.sub.2],[s.sub.1], [s.sub.2],[s.sub.3],[s.sub.1],[s.sub.2],[s.sub.3], [s.sub.4],.... Let M > 1 be an upper bound of the double sequence ([absolute value of [w.sub.u,v]]). Further, let ([[member of].sub.n]) be a strictly monotonic null sequence.

Since C(w) = D(w), [t.sub.1] can be written in the form [t.sub.1] = [q.sub.11][l.sub.11] + [q.sub.12][l.sub.12] + ... + [q.sub.1n(1)][l.sub.1n(1)] where [q.sub.1i] > 0 for all i, [[summation].sup.n(1).sub.i=1] [q.sub.1i] = 1, and [l.sub.11],[l.sub.12], ..., [l.sub.1n(1)] [member of] L(w). By (1) thru (5) and (S) there exist positive integers [m.sub.1] and [r.sub.1], both greater than 1 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next, there exist [q.sub.21],[q.sub.22], ... q.sub.2n(2)] where [q.sub.2i] > 0 for all i, [[summation].sup.n(2).sub.i=1] [q.sub.2i] = 1, and [l.sub.21].[l.sub.22], ... [l.sub.2n(2)] in L(w) such that [t.sub.2] = [[summation].sup.n(2).sub.i=1] [q.sub.2i][l.sub.2i]. By (1) through (5) of Theorem 2.6 and (S) there exist positive integers [m.sub.2],[r.sub.2] with [m.sub.2] > [m.sub.1] and [r.sub.2] > [r.sub.1] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In the above and in what follows, if t and s are positive integers, with t > s let

[L.sub.st] = (NxN) \ ({(u,v) : u > t or v > t} [universal] {(u,v) : u,v < s}).

Continuing, as in the first two steps, we obtain two strictly increasing sequences ([m.sub.k]) and ([r.sub.k]) of positive integers satisfying:

(A) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(B) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(C) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(D) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [t.sub.k] = [[summation].sup.n(k).sub.i=1] [q.sub.ki][l.sub.ki] [q.sub.ki] > 0 for all i, [[summation].sup.n(k).sub.i=1] [q.sub.ki] = 1, and [l.sub.k1],[l.sub.k2], ... [l.sub.kn(k)] in L(w).

We now construct the subsequence z of w eluded to in the statement of the theorem. To achieve this consider the sequence of sets:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ... where [L.sub.1] = {(u,v) : u,v [less than or equal to] [r.sub.1] - 1). Notice that these sets are pairwise disjoint and their union is NxN. We now proceed to partition these sets into n(l), n(2),n(3), ... sets respectively using the linear ordering in Definition 2.3, namely we consider [z.sub.1] < [z.sub.2] < [Z.sub.3]....

Start with [L.sub.1]. We partition it into n(1) pairwise disjoint subsets having union [L.sub.1]. Denote by < the linear ordering of NxN from Definition 2.3. Let [x.sub.11] be the first positive integer satisfying:

a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [x.sub.12], [x.sub.12] > [x.sub.11] be the first positive integer satisfying:

b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Continue, in order, defining the strictly increasing sequence [x.sub.11], [x.sub.12], ..., [x.sub.1n(1)] satisfying

c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

d) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [w.sub.1] is the last element in [L.sub.1] with the ordering again being the one from Definition 2.3. For each k, k = 2,3, ..., in a similar way, we partition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Namely, there exist [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that are strictly increasing, such that

e) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

f) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

g) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[omega].sub.k] is the last element in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now a) through g) are easy consequences of ([beta]), ([gamma]), (B) and (C). And now, finally, after having achieved the partitions of I4, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ..., we construct the required subsequence z of w. We define in pieces, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ... respectively. Namely we can construct a subsequence z of w (see the definition of a subsequence) such that:

[absolute value of [z.sub.uv] - [l.sub.11]] < [[epsilon].sub.1]/[C.sub.1][2.sup.j]

where (u,v) is the j-th element in [L.sub.1] such that (u,v) [less than or equal to] [x.sub.11],

[absolute value of [z.sub.uv] - [l.sub.12]] < [[epsilon].sub.1]/[C.sub.1][2.sup.j]

where (u,v) is the j-th element in [L.sub.1] such that [x.sub.11] < (u,v) [less than or equal to] [x.sub.12], ...

[absolute value of [z.sub.uv] - [l.sub.1,n(1)-1]] < [[epsilon].sub.1]/[C.sub.1][2.sup.j]

where (u,v) is the j-th element in [L.sub.1] such that [x.sub.1,n(1)-2] < (u,v) [less than or equal to] [x.sub.n,1(1)-1],

[absolute value of [z.sub.uv] - [l.sub.1,n(1)] < [[epsilon].sub.1]/[C.sub.1][2.sup.j]

where (u,v) is the /-th element in [L.sub.4] such that [x.sub.1,n(1)-1] < (u,v) [less than or equal to] [[omega].sub.1] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (by (5) in Theorem 2.6).

Now also, for each k, k = 2,3, ... [z.sub.uv] is defined on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that:

[absolute value of [z.sub.uv] - [l.sub.k1]] < [[epsilon].sub.k]/[C.sub.k][2.sup.j]

where (u,v) is the j-th element in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ...

[absolute value of [z.sub.uv] - [l.sub.ki]] < [[epsilon].sub.k]/[C.sub.k][2.sup.j]

where (u,v) is the j-th element in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[absolute value of [z.sub.uv] - [l.sub.k,n(i)]] < [[epsilon].sub.k]/[C.sub.k][2.sup.j]

where (u,v) is the j-th element in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let us simplify notation by setting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By the definition of a subsequence and the fact that each [l.sub.ki] is a limit point of w, a subsequence satisfying all of the above inequalities exists.

Now we show that such a subsequence z satisfies the required conditions, namely each t [member of] C(w) is a limit point of Az. To see this, let k be a positive integer, then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that the second term on the right hand side of [(xx).sub.k] is taken to be zero in case when k = 1. Let us examine the first term on the right hand side of [(xx).sub.k], denote it [(xxx).sub.k]. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where here [z.sub.uv] = [l.sub.ki] + [[gamma].sub.uv] if (u,v) [member of] [B.sub.ki], i = 1,2, ..., n(k) and [B.sub.ki], i = 1,2, ..., n(k) are the n(k) sets into which [L.sub.k] has been partitioned. The first set of parentheses, inside the absolute value signs, on the right side of [(xxx).sub.k] has absolute value less than 1/[10.sup.k]. The second set of parentheses, inside the absolute value signs, on the right side of [(xxx).sub.k] has absolute value less than [[member of].sub.k].

Finally, the second and third terms on the right hand side of [(xx).sub.k] satisfy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so that [(xx).sub.k] < 1/[10.sup.k] + [3[[epsilon].sub.k] for each k, which shows that each [t.sub.k], and consequently each [s.sub.k] is a limit point of Az. Since {[s.sub.n] : n [member of] N} is dense in C(w), each t [member of] C(w) is a limit point of Az.

We finish, by stating a theorem that is the exact analogue of Theorem 3.2 in [9], and whose proof follows that of the mentioned theorem.

Theorem 3.2. If A = ([a.sub.m,n,u,v]) is a four dimensional bounded regular summability matrix satisfying (S) and w is a bounded double sequence, then there exists a rearrangement zofw such that each t in the Pringsheim core ofw is a P-limit point of Az.

References

[1] R.P. Agnew, Summability of subsequences, Bull. Amer. Math. Soc, 50 (1944), 596-598.

[2] R. C. Buck, A note on subsequences, Bull. Amer. Math. Soc.,49 (1943), 898-899.

[3] J. A. Fridy, Summability of rearrangements of sequences, Math. Z.,143 (1975), 187-192.

[4] J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301-313.

[5] J. A. Fridy and C. Orhan, Statistical limit superior and limit inferior, Proc. Amer. Math. Soc, 125 (12) (1997), 3625-3631.

[6] H.J. Hamilton, Transformations of multiple sequences, Duke. Math. Jour., 2 (1936), 29-60.

[7] K. Knopp, Zur Theorie der Limiteurung sver Fahren (Erste Mitteilung), Math.Zeit.,31 (1930)

[8] H.I. Miller, Summability of subsequences, rearrangements and stretchings of sequences, Akad. Nauka i Umjet. BiH. Rad. Odjelj. Prirod. Mat. Nauka, 19 (1980), 95-102.

[9] H. I. Miller and R. F. Patterson, Core theorems for double sequences and rearrangements, Acta Math. Hungar., (2008)--online, 10 pages.

[10] R. F. Patterson, Analogues of some fundamental theorems of summability theory, Internat. Jour. Math, and Math. Sci., 23 (1) (2000), 1-9.

[11] R. F. Patterson, Characterization for the limit points of stretched double sequences, Demonstratio. Math., 35 (1) (2002), 103-109.

[12] R. F. Patterson, X-rearrangements characterization of double sequences Pringsheim limit points, in press.

[13] F. Moritz and B.E. Rhoades, Almost convergence of double sequence and strong regularity of summability matrices, Math. Proc. Camb. Phil. Soc, 104 (1988), 283-294.

[14] G. M. Robinson, Divergent double sequences and series, Amer. Math. Soc. Trans., 28 (1926), 50-73.

[15] L. L. Silverman, On the definition of the sum of a divergent series, unpublished thesis, University of Missouri Studies, Mathematics.

[16] O. Toeplitz, Uber Allgenmeine linear mittelbrildungen, Prace Matemalyczno Fizyczne (Warsaw), 22 (1911).

* The research presented in this paper was partially funded by a grant from the Ministry of Education and Science, Kanton Sarajevo

Received by the editors December 2009.

Communicated by F. Brackx.

2000 Mathematics Subject Classification : 40D25, 40G99, 28A12.

Faculty of Engineering and Natural Sciences

International University of Sarajevo

Sarajevo, 71000

Bosnia-Herzegovina

E-mail: himiller@hotmail.com, lejla.miller@yahoo.com
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Author:Miller, Harry I.; Wieren, Leila Miller-Van
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
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Date:May 1, 2011
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