AN ORDERED SET OF NORLUND MEANS

Norlund methods of s-m.bility are studied as mappings from i into i" Those Norlund methods that map 41 into i are characterized. Inclusion results are given and a class of Norlund methods is shown to form an ordered abelian semi-

Let N denote the collection of all such Norlund methods N If P + 0 for all p n n >= 0, that is K 0, then Pn_k/Pn, if k <__ n and Np[n,k] 0 otherwise.
Throughout we let P0 if 0 < n < K and P if n > K.The N transform of n n n p a sequence x is then given by NpX, where n (NpX) n (i/n) kZffi0Pn_kXk for all n > O.
A matrix summability method is called an -E method if and only if it maps the space I -= into itself.In [4], Knopp and Lorentz proved that the matrix method A isif and only if there exists some M > 0 such that sup Z lank < M. k n=0 In the special case of a Norlund method N we have: P THEOREH 1.The Norlund method N isif and only if P (i) p e J, and (ii) -+ 0 as n / =. n PROOF.First suppose that (i) and (ii) hold.Since p it implies lira exists and by (ii) is non-zero.So there exist strictly positive numbers n n H and such that rIpnl<H and lnl > for all n > 0. Thus for each fixed k, Thus by the Knopp-Lorentz Theorem N is -.P Now suppose that N is -.Then P r.IPn_k/'nl. 0(i).n=k In particular if n=k it implies [1/n] 0(i) and hence @ 0 as n /-.n Now suppose that p .We assert that the series l( 1-SN/SN+m.But SN+m + as m / =.So choose m large enough such that SN+m > 2S.N. e= Z (Ip.I/li' I) > " n .The theorem now follows.
n=N+l COROLLARY 1 [i, Theorem 4].Let N be a Norlund method with pn > 0 P0 > 0. p Then N isif and only if p e .P 2. We make the following definitions.
DEFINITION.Let N denote the collection of all N that are E-methods.E p DEFINITION.Let E(Np) consist of all sequences x such that NpX is in E.
and N N is k-stronger than N DEFINITION.Given two Norlund methods Np q q P if and only if E(Np)C_ E(Nq).The method Nq is strictly k-stronger than Np pro- vided (Np) E(Nq), and Np and Nq are k-equlvalent provided (Np) (Nq).
DEFINITION.Given N and N define formally: The next propositions follow by an argument similar to the one used in Theorem 18 of [3]   n n PROPOSITION 1.If N E' then the series r. pn z and r. (iii) qn bnP0 + + b0Pn' and (iv) n bn'O + + bon PROPOSITION 3. SuPpose N v E and the sequence S {S } is in (Np).Then p n the series S(z) r. S z n has positive radius of convergence.
n n PROOF Let h(z) I/p(z).By Proposition I, p(z) defines an analytic function for z[<L Since Po + 0, by the continuity of p(z) there exists some a e (0,i) such that p(z) 0 for all z e (-a,a).Therefore h(z)  3. The symmetric product g p*q of the sequences p and q is defined by gn P0qn + + Pnq0 for all n > 0. Given Norlund methods N and N in N we p q say N N is the symmetric product of N and N provided g P*q p q In order to prove an inclusion result for two Norlund methods in N we need the following lemma.
Lemma i.Let the complex sequences p and q be given and define r p*q.
Suppose Np, Nr e h/.Then in order that (Np) C_ (Nr) it is necessary and sufficient that there is some M > 0, independent of k, such that [kl Z ]qn_k/n[ < M.
Proof.Let x {x } be any sequence.Then n 5. DEFRANZA Let enk-qn_kPk/R n if k <__ n, and 0 if k > n.Then by the Knopp-Lorentz Theorem '(Np) (N r) if and only if there exists some M > O such that sup { .leak } < 14.
The lemma can now be used to get the desired inclusion result.
Proof.In the previous 1emma replace the sequence q with the sequence b which implies r-= p*b q.Then (Np)_ (Nr) (Nq) if and only if there exists some M > 0, ndependent of k, such that [k[.Z.i bn_k/ n < 14" But since Np, Nq , lkl and lkl respectively are bounded by two strictly positive constants.Thus [k[ Y. Ibn_k/n < M independent of k, is equivalent to b4. and Corollary 2. Suppose Np, Nq 4" The (1) 4(Np) 4(Nq) if and only if both a {an } 4 and b {bn } 4 (ll) 4(Np)4(Nq) if and only if a {an } 4 and b {bn } 4.
(Np (Nq) 1/nk, Moreover by Corollary 3, If Yn k > 1 and n > 0, then (Np) (Ny). 4. We now show that N forms an ordered abelian semigroup.The order relation is set inclusion between the absolute summabillty fields, and the binary operation is the symmetric product of the generating sequences.We need the following Lemma 2. Suppose N e N and p is a sequence for which lim P P + 0. q n-= n Let r p*q.Then llm (R_/PQ n),, 1. n.-o Proof.We assert that N is a regular method.By the Silverman-Toeplltz q Theorem, see for example [6], it suffices to show that /(1) qn-9 Qn + 0 as n--for each fixed > 0, and n --0 But since N e , (1) and (ii) follow.Now for all n sufficiently large q n n n/n (iI n) 1,.. z__O rk (iI n) kXoqn_k'k which is the N transform of the convergent sequence {P }.Thus q n Lemma 3. If N Nq e N and r p'q, then N e N.

2 .
If Np, Nq N then the series a(z) n r anZ and b(z) r. b z n have positive radii of convergence and, , it follows that the series I Pk(NpS)kzk converges for Therefore Z S z n has positive radius of convergence since n n E S z n {h(z)} {k.0k(NpS)kzk}.
n=0 n Lemma i, we have lim + O.In order to have N e , we n r n need to show N is an method.By Theorem I, it suffices to show that r 4.