A converse product theorem in summability

ANNALES SOCIETATIS MATHEMAT1CAE POLONAE Series I: C O M M ANTATIONES MATHEMATICAE XXII (1980) ROCZNIKI POLSK IEG O TOWARZYSTWA M ATEM ATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXII (1980)

M^angalam

R.

(Winnipeg, Canada)

A converse product theorem in summability

1. Introduction. If A is a summability method applicable to (some) sequences and if В is a sequence-to-sequence transformation, then the product method A ■ В is defined as the iteration product A(B) of the methods A and B;

i.e. A - В sums a sequence s = {s„} if and only if A sums the sequence Bs.

(If A and В are matrix methods, it may happen that A ■ В and С = AB behave differently from each other.) Product theorems of the type A

A

В (sometimes restricted to a given class of sequences) have been studied by Szâsz [10], Jakimovski [2], Rajagopal [7], Ramanujan [8] and others (including earlier workers like Zygmund [12] who proved some special product theorems). Converse product theorems of the type A ■ В c A were first attempted by the author in [4]. In the present note we give (in Theorem 1) an elementary converse product theorem for the product of Euler-Knopp and Cesàro methods for bounded sequences and in Theorem 2 we prove a generalization which identifies precisely why the equivalence relation in Theorem 1 holds; at the same time, Theorem 2 is more general than another result quoted as Lemma 1 below.

2. Definitions and notation. The Euler-Knopp methods Ep (0 < p < 1), the Borel method B, the Taylor methods Tp (0 < p < 1) and the Meyer-Konig methods Sa (0 < a < 1) constitute the family Г of circle methods (“Kreis- verfahren”) which are regular and not equivalent to convergence. These methods were studied comprehensively for the first time by Meyer-Konig [3].

For definitions and further details of these and of the Cesàro and other Hausdorff methods, see also Hardy [1], Ramanujan [9] and Zeller and Beekmann [11].

3. Lemmas and theorems.

Lemma 1

(Ramanujan

Parameswaran

Every conservative Haus­

dorff method H = (H, p„) with lim pn = 0 sums all Borel-summable bounded

sequences.

132

C

. I f 0 < p < 1 and r > 0, then Ep a Cr for bounded sequences.

(This is well known otherwise also.)

L

2 (Parameswaran [5]). The conservative method ( H, p„) will sum no Borel summable bounded divergent sequences if lim p„ Ф 0.

T

1. Let 0 < a < 1 and p, r > 0. Then (1) Ex Cr ~ Cp for bounded sequences.

This can be stated also in either of the equivalent forms

(2) (а) Ея Cr c: Cr or (b) Ex Cr ^ C, (for bounded sequences).

P roof. As usual, let (c) and (m) denote the sets of all convergent, and bounded, sequences respectively. If se(m) and Ex Crse(c), then by Lemma 1 (since Ex c= J3), Ci Cr se(c) and hence C 1+rse(c). Since se(m), this implies that Cpse(c). Thus we have Ex Cr c Cp ~ Cr c Ex Cr for bounded sequences and (1) is proved. (Note that if we consider only bounded sequences and conservative matrices A and B, then AB = A ■ В .) The last part of the theorem is now obvious.

T

2. Let H — ( H , p„) be a conservative Hausdorff matrix method and let A be any member of the family Г defined in Section 2. Then the methods A ■ H and H are equivalent for bounded sequences if and only if lim pn = 0. In particular, for 0 < a < 1,

EaH cz H

(and hence EaH % H) for bounded sequences if and only if lim pn = 0.

P ro o f. If H is a conservative Hausdorff matrix and se(m ), then Hse(m).

Since the members of Г are all equivalent for bounded sequences (Meyer- Konig [3], Satz 25), it is enough to prove the theorem when A is an Euler method Ex (0 < a < 1).

(i) Sufficiency of the condition. Let H — (H, pn) = (hn k) be a conservative Hausdorff matrix with lim pn = 0, and let Exu = ExHs be convergent, where s is a bounded sequence and 0 < a < 1. By the well-known Tauberian theorem for £ a, it is enough to prove that

um ~ un Q as m > n 00> ( m - n ) n ~ ll2-+ 0.

Now there exists a function g ( t ) e B V [ 0 ,1 ] such that g(\) = g ( l — 0) and K k

tk( l - t f ~ kdg(t)

о and и = Hs is given by

un = É sk ^(k)tk( l - t f ~ kdg(t) = X sk je„'k(t)dg(t),

k=0 О k

Converse product theorem 133

where enk(t) = (£)rk(l — t)n k. Then

m 1

um- u n = X {emtk(t)-e„yk(t)}dg(t),

к = О 0

where we take eiyj{t) = 0 if j > i. So if {s„} is bounded,

1 m

(1) um- u„ = 0{$\dg(t)\ £ \em,k( t ) - e Htk{t)|}.

0 fc= 0

Since g(t) is continuous at t = 1, the contribution to (1) of the interval [1 — h, 1] can be made arbitrarily small by proper choice of h < 1. It is therefore enough to show that, for each fixed h with 0 < h < 1,

m

(2) £ km,k(0-e„,fc(f)I -» 0 uniformly in 0 < f < 1 -Л

k — 0

as m > n -> oo, (m — n)n~ll2~* 0.

Now let 5 denote the operator defined by <5w„ = u„ — when more than one variable suffix occurs, we add a suffix to Ô to denote which variable is operated on. Since

i = n+ 1

the assertion (2) will follow if we prove that П

(3) Y

\S„e„yk(t)\ = 0 ( n ~ 112) uniformly in 0 ^ t ^ 1— h.

k = о

Now ônenk(t) = —tdken- lyk(t) and so the expression on the left in (3) is equal to

№ i £ i < w , . a o i .

k = 0

For fixed n, the numbers en- 1 yk(t) increase to a maximum and then decrease;

so the sum in (4) does not exceed twice this maximum. But uniformly in 0 < t < 1 (though it suffices for us if it holds uniformly in 0 < t ^ 1 — h),

omax/ n ,k(0 = 0 ( { n t { \ - t ) } ~ m ).

Thus the quantity in (4) is O (f1/2 (1 —1)~1,2 n~ 1/2} and the desired result (3) follows. This completes the proof of the sufficiency part of the theorem.

The author wishes to thank Professor B. Kuttner for pointing out an error in the first version of the author’s proof and for suggesting the present version.

(ii) Necessity of the condition. Let H — (H, gn) be any conservative Hausdorff

matrix with lim gn Ф 0. We take s to be any Borel-summable bounded

134 M. R. P a r a m e s w a r a n

divergent sequence; then of course, Epse(c) for 0 < p < 1 and hence EpHs — HEpse{c). But H s^(c), since no conservative Hausdorff matrix with lim iin Ф 0 can sum s, by Lemma 2. Thus, for every such matrix H we have EpH Ф H for bounded sequences.

References

[1] G. H. H ardy, Divergent series, Oxford 1949.

[2] A. J a k im o v s k i, Some relations between the methods of summability o f Abel, Borel, Cesàro, Holder and Hausdorff, J. d’Analyse Math. 3 (1953/54), p. 346-381.

[3] W. M e y e r -K d n ig , Untersuchungen iiber einige verwandte Limitierungsverfahren, Math.

Zeit. 52 (1949), p. 257-304.

[4] M. R. P a ra m esw a ra n , Some product theorems in summability, ibidem 68 (1957), p. 19-26.

[5] — Some remarks on Borel summability, Quart. J. Math. (Oxford) (2) 10 (1959), p. 224-229.

[6] — Some new product theorems in summability (to appear).

[7] С. T. Raj a g o pal, Theorems on the product o f two summability methods, with applications, J. Indian Math. Soc. (NS) 18 (1954), p. 89-105.

[8] M. S. R a m a n u ja n , Theorems on the product of quasi-Hausdorff and Abel transforms, Math. Zeit. 65 (1956), p. 442-447.

[9] — On Hausdorff and quasi-Hausdorff methods of summability, Quart. J. Math. (Oxford) (2) 8 (1957), p. 197-213.

[10] O. S zâsz, On the product of two summability methods, Ann. Soc. Polon. Math. 25 (1952/53), p. 75-84.

[11] K. Z e lle r and W. B eek m a n n , Theorie der Limitierungsverfahren, Springer, Berlin- Gottingen-Heidelberg 1970.

[12] A. Z y g m u n d , Remarque sur la sommabilité des séries de fonctions orthogonales, Bull.

Internat. Acad. Polon. Sci. Lettres, Cl. Sci. Math. Nat., Ser. A (1926), p. 185-191.

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