ON IIESZ SIIMMABILITY ACTOBS I.J. MADDOX. (Received August 17, 1962)

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1 ON IIESZ SIIMMABILITY ACTOBS I.J. MADDOX (Received August 17, 1962) 1. In this note we give, in Theorem A, necessary and sufficient conditions for a sequence of real numbers (fin) to be such that ann is absolutely convergent whenever fan is summable (R, K), K>0. The case K=0, i.e. (R, X, 0) equivalent to convergence, was dealt with by Fekete [3]. If we take Xn=n in Theorem A, then the known equivalence of (R, n, K) and (C, K) summability yields a result due to Bosanquet [1], Theorem 3 (the case p = 0, K>0). Jurkat [6], Satz 1, has given a matrix method of summability which, with certain restrictions on Xn, is equivalent to (R, X, K) summability for all K>0. We shall employ this method of summability to give a necessary condition for anon<oo whenever tan is summable (R, X, K). In what follows we shall refer to a number of summability methods. (i) Suppose that [X] nis infinity. Define for K>0, a sequence of non-negative numbers increasing to If or AK (w)-s (co-co), then we say that fan is summable (R, X, k) to s. (ii) If in (i) above w takes only the values Xn+1, then we say fan is summable (R, X, rc) to s. For 0<k c 1, Jurkat [4], Satz 2, has shown that (R, X, c) and (R, X, c) summability are equivalent. (iii) Let p>0 be an integer and k j p+8, 0<0 1. Define Successive applications of this last formula with B=1define C a further application defines C. If CK [sm]/ck[1]s(n-oo), then we say fan is summable CK to s. (iv) We say fan is summable B j if I ztn<oo,

2 432 I. J. MADDOX if (v) Let R (co) be defined as in (i). Then f an is summable R, X, x, k 0, By I R, 0 summability we mean an <oo. 2. For the proof of Theorem A we shall require some preliminary lemmas. LEMMA 1. Let k=p+o (p>0 integral, 0<B c 1), and suppose that An=-Xn) increases. For non-integral ic let n be monotonic and n (a,n increase. Then the following are equivalent This was proved by Jurkat [6], Satz 1. LEMMA 2. If AK(w)=o(w), K>0, then Sn=am=o(An), An=-Xn. Lemma 2 is the limitation theorem for Riesz means (see for example Hardy and Riesz [4], Theorem 22). LEMMA 3. Let [U} nconverge, j an j<0o and an=sn,mum (n=0,1,...). Then/T I Sn.n I<00 This follows from a result due to Chow [2], Lemma 6. LEMMA 4. Let A and B be normal matrices. If IanEn is summable (B j, whenever San is summable A, then PROOF. Let un=an, mam, to-, bn, mamcem, and an=to-t1. Then fun

3 RISZ SUMMABILITY FACTORS 433 is convergent, and =bn, nansn>(bn,m-bn -1, m)amem Sn, mum, sn, n=bn, na n, n4sn, sn, m=bn, na'n, mn+er(bn, r-bn-1, r)ar, m for 0<m<n-1. Here (ar) is the reciprocal matrix obtained by solving for an in terms of un. Since an, n=a, we have on applying Lemma 3, bn, nan, nt5n I>sn, n I<00 This proves the lemma. 3. We now prove the main result. THEOREM A. Suppose that the conditions of Lemma 1 are satisfied. Then ZanS is absolutely convergent whenever an is summable (R, X,,c), k>0, if and only if <00, An=Xn+1/(Xn+1-n) PROOF. Necessity. By Lemma 1 we may employ dc summability in place of (R, X, rc) summability. For positive integers p, we have Hence It is readily shown by induction on r, that for integers r >1, (1) =(Xn+r-Xn).-X)/r. Thus it follows from (1) that

4 434 I. J. MADDOX x, n=un, mam, Hence we have dn, n=(x1 n+-x rf(ic+1) if 0<k c 1, and dnan=(x+1 n-xy(xn+1-x). (x+2-x)/(p+1)i(b+1) if,c>1. Let us now consider the CK transform of fan C/CK f ll=>anmam,. an, n=d/ck [11. Since anon <co, whenever fan is summable CC, we have by Lemma 4 that (Z) Now it was shown by Jurkat in the proof of Lemma 1 that a,a constants. Also since A.n increases we have din 0(1)dn11. Hence for r=2,3,p+1, xn+r-xn i I dxn+l I I DXn+r-1 Xn+l-1,n y I n I I dxn I (4) (x+1 np-xn)-xn}=0(1)(xn+1-n)p. Thus by (2), (3) and (4) we have This proves the necessity. We note that the above proof can be used to establish a rather stronger result. For let us take, in Lemma 4, bn,m=(1- >o. Then if 4ann is summable ( R, X, it is summable B. Hence by Lemma 4, By (2), (3), (4) and (5) we then have (6)

5 RIESZ SUMMABILITY FACTORS 435 Thus (6) is necessary foran to be summable R, h, 0, whenever fan is summable Cx. We note that when 0<K c1, no restriction on Xn is required, since CK is equivalent to (R, X, K), which is equivalent to (R, X, K) by an earlier remark (Section 1,(ii)). Sufficiency. Suppose thatan n<co. Since fan is summable (R, X, K) and An increases, we have by Lemma 2, Hence an=0(a+an-1)=o(an). This proves the theorem. REFERENCES [1] L. S. BOSANQUET, Note on convergence and summability factors, Journ. l London Math. Soc., 20 (1945), [2] H. C. CHOW, Note on convergence and summability factors, Journ, l London Math. Soc., 29 (1945), [3] M. FEKETE, Summabilitasi factor-sorozatok, Math, es Termes. Ert., 35(1917), [4] G. H. HARDY AND M. RIESZ, The General Theory of Dirichlet's series, Cambridge Tract No. 18, [5] W. JURKAT, Uber Rieszsche Mittel mit unstetigem Parameter, Math. Zeit., 55(1951), [6] W. JURKAT, Uber Rieszsche Mittel and verwandte Klassen von Matrixtransformationen, Math. Zeit., 57 (1953), DEPARTMENT OF MATHEMATICS, THE UNIVERSITY LEEDS 2, ENGLAND