Proc. Indian Acad. Sci. (Math. Sci.), Vot. 104, No. 2, May 1994, pp. 373-376. Printed in India.

Rearrangements of bounded variation sequences

MEHMET ALI SARIGOL Department of Mathematics, Erciyes University, Kayseri 38039, Turkey

MS received 1 March 1993; revised 19 November 1993

Abstract. Let by be the set of all bounded variation sequences. In the present paper we deduce from a theorem of Meat's a necessary and sufficient condition for the rearrangement (a~t ~) to be of bounded variation whenever (at)~ by; interestingly it coincides with Pleasants' criterion for convergenoe-preserving.

geywer&t. Rearrangements; sequences.

Let Y-a~ be an infinite series of real numbers and p be a permutation of N, the set of all positive integers. The series Y-a~k~ is then called a rearrangement of Y~ak. A classical theorem of Riemann states that if Y.a~ is a conditionally convergent series and s is any fixed real number (or 5- oo), then there is a permutation p such that ]ga~k~ = s. Thus it leads us to the problem of characterizing the rearrangements which do not change the sum or the convergence or even the divergence of the series. They were studied in [1]-[9] and by others. Of special interest is a paper by Pleasants [5] giving a characterization of permutations which transform convergent sequences to convergent sequences. In this paper we consider questions similar to those above, but for rearrangements of bounded variation sequences. We recall some notation before stating the precise problem.

DEFINITION Let y denote the set of all convergent series of real numbers. A permutation p on positive integers is then called convergence-preserving (CP for short) if ap = (a~t~)~y for any sequence a = (ak)ey, 15]. We shall denote the finite consecutive run of integers i,i+ 1..... j-1,j by [i,j] and we shall call such a set a block. Every finite set F of N is a union of disjoint, non-adjacent blocks of consecutive integers. Let v(F) denote the number of such blocks and p-1 be the inverse of p. With this terminology, the result of Pleasants on CP permutations can be stated as follows.

Theorem 1. [5; p. 135]. A permutation p is CP if and only if there is a constant M (= M(p)) such that v(p -1 {1,2 ..... k}) <~ M for all keN. We now give the following notation similar to the above definition, for sequences of bounded variation. 373 374 Mehmet All Sari061

DEFINITION Let p be a permutation of N. We say that p is bounded variation-preserving (written BVP) if the rearrangement induced by p, ap = (a~k)), is of bounded variation for all a = (ak)~bv. It may be noticed that, if a~bv then any rearrangement of a is not necessarily of bounded variation. For example, take a=(1,1/2, .... 1In .... )~bv and define permutation p as

f 2n/3, if n = O(rnod 3) p(n)= ~(4n- 1)/3, ifn~_ l(mod3) [(4n + 1)/3, if n- 2(mod3). Then (a~,))~bv, since

[a~.~ -- a~._ t)l > ~ 3(2n + l)/2(n -- 1)(4n -- 1) = oo. .=2 .m 1 (mod3) In this paper we give the following characterization of such permutations and obtain a relation between CP and BVP permutations.

Theorem 2. A permutation p is BVP if and only if there exists a positive integer M such that v(p -1 {1,2 ..... k}) <<. M for all k. We deduce this from the following result of Meats [5] on bounded variation sequences.

"r~orem 3. The infinite matrix A = (a~ ) transforms every x e bv into ( ( Ax),)ebv, where (Ax). = Za~xk if and only if

(i) ~ a~ converges for each n, and k=l (ii) There exists a positive constant M such that, for all k,

(a i -- a._ l,l) <<. M. n=l i

Proof of Theorem 2

Suppose that p is any permutation on N and xebv. A rearrangement of x by p can be considered as a matrix transformation in the following way. Set a~ = 1 if k = p(n), and a~ = 0 ifk ~ p(n). Then (Ax), = x~,), i.e., Ax = (x~,)), the rearrangement produced by p. Since p is one-to-one and onto mapping, each column and each row of (a~) contains exactly one nonzero term. Thus for each n, ~ a~ = 1 and also for each k > 1, k=l I- 1 a._,,,) n=l ri=k n=l i=I Since p-l is also the permutation on N, le{p- 1(1),p-1(2), .... p-l(k)} for all sufficiently large k. Now arranging the intergers in {p-l(1) ..... p-a(k)} in increasing order, we Rearrangements of bounded variation sequences 375 write the same set in the following form:

{ 1 = r~ ), 2 .... , t~', r~ff', r~ ' + 1..... t]t,, taf), r~ff' + 1..... t~ff),..., r~ ', ta~' + 1.... , t~'}, (1) where r~' ~ t~' < r~ ,- 1 < t]" < r~ ,- 1 <... < r~' - 1 < t~'.

It is clear that v(p-x [1, k])=jr + 1. Because of the definition of p, there is only one nonzero term in each row corresponding to the integers in (1), and so

a.~= {1, if r k', j=O, 1..... Jk ~=: O, otherwise, which implies that k f-l, if n=t~ tJ ~l(a'~-a'-:")= 1 I, if n=r~ ~), j = O, 1.... jr. = O, otherwise Therefore it follows that

Z (a,~ - a._ - 2(jk + 1), s=l i=1 because the above sum is '1' added to itself 2~jt-4-1) times. So, by Theorem 3, Ax = ((Ax),)~bv iff there is a constant M such that jt ~ M for all k. This completes the proof.

COROLLARY 4 A permutation is CP if and only if it is BVP. In [5] it is shown that there exists permutations p which are CP but the inverse p-: not CP. In particular it follows that.

COROLLARY 5 The set of all BVP permutations does not form a group.

Acknowledgement

The author expresses his gratitude to the referee for valuable suggestions. This research was supported by TBAG-(~G2 (TOBITAK).

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