負数次のヘルダー総和法について （英文）
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(2) Vol. 10, No. 2 Journal of Hokkaido Gakugei University Dec. 1959. On the Holder Mean of Negative Order (I) Shiroji MJURA The Stzidy of M'athematies, Hakodate Branch,. Hokkaido Gakugei U'niversity. ^?Nt6?^: ii^®-'^^-Wi^R:o(^r. 1. Introduction. Amnon Jakimovski and C. T. Rajagopal proved the following Tauberian theorem for the Holder mean of negative order —k in [I], [2]. Theorem of Jakimovski. Let k be a fixed positive integer. A necessary and sufficient. condition for {s»} to be summable (H, —k) to s is that {&„} should be summable (A) to s and \im(n}-^Sn-^ =0. n ->^>. Theorem of C. T. Rajagopal. (a) If (i) {s,J is summable (A) to s, and if (ii) for a positive integer k, nk^/CSn-/c==0, %->oo, then {s,J is summable {H, —k) to s.. (b) Conditions (i) and (ii) are also necessary for {s,J to be summable {H, —k) to s. In this paper, we will extend the above result to the case of some fixed positive fractional number k. 2. An extension of the above theorem. By the application of the difference of fractional order defined in [3], we will make an extension of the above theorems in the case of a positive fractional number k. Theorem 1. If {Sn} is summable (A) to s and if, for a fractional number k, 0<k<l, nkS "(s,,)l) - 0(1), n^oo, then {5,1} is summable (H, —k) to s, where Snz=ao+a^+---+an.. Proof. Let us now put k^=—{S. From our assumption Q<k<l, we have —1<<9<0.. Then our hypothesis is that n-s SB (s,,) -= 0(1), M->oo (/9> -1), that is, (S8(s,,)/A,e).(A,8/n8) - 0(1\ n-^oo. 1) The difference of fractional order in [3] is defined by the operation of "inverse summation" : S- (s») = 2 A^sy, 1^=0. which reduces, in the case k=l, to S (s,i) = Sn—S,i-l,. provided we write s-i=0. Writing S/i(sn) = 2 Ak^_\sv, we have, for all real a, ft Sct {SB (sn)} 1/=0. =Sa+S(s»), in particular S-a {Sa (s»)}=S°(sn)=s». Also Sl(s»)=So+Si+---+s», and ^=(aJ;w)= 92'. (-l)n(-a,-l)=Sff(l)..
(3) On the Holder Mean of Negative Order Since we usually write. Sy(sj. cs(^)-=-"-^-a3>-i). "AT. for the w-th Cesaro mean of order /3 of a sequence s,/ (defined for ^-=0,1, 2,---), we get Cs (s») -A^- = 0(1), n-^oo Q9> -1). nB. From Stirling's Theorem, as is well known, we have. A°--a-r^)+P.C-ts-u+^^ and it follows that A,6 1 lim -^n6 ~~ F(0+l) • Therefore Cs(5,,)-=0(l), n^oo (/3>-1). Hence, s«-0(l) (C, <9), %-^oo (^9>-1), that is,. 20 an=Q{C, ^)03>-1).. ?l=0. Applying the theorem proved by A. F. Anderson in [4]2), if. then 2A^(fl,,-fl,,+i)=0(C, ^+1); that is, we have n=0. S(%+1)(^-^)-0(C, 0+1).. 71=0. Taking the n-th partial sum of the series Z;(%+1) (an—an+i), we obtain n=o. n 2 (v+l)(a,,-av+i) -- (<3o-ffi)+2(fli-a2)+•••+(%+!) (^-On+i). I/=0. -^ ffo+(?i+---+fln— (7t+l)a,t+i. - Sn-(n+l)an+i.. Hence, we get s,,-(n+l)(3!,.+i = 0(1)(C, j9+l), %-)>oo.. From 2 On = 0 (C, j9) 09>—1) and the inclusion theorem for Cesaro means, we have. So»=0(C, ^+1)(/5>-1),. n-0. that is Sn = 0(1) (C, /3+1), %-^00. Therefore we have (%+l)ff,^i = 0(1)(C, ,3+1), n^oo, 2) Theorem of Anderson. For P real (P^O), k>-l, if S Ap-l a» is summable (C, fc), then 2 A^ (a,,-an+i) is summable n. (C, k+1) with the same sum.. =. 0. n. =. 0.
(4) Shiroji Miura that is, nan = 0(1)(C, 13+1), n->w. As <3>—1, we get nan = 0(1) (77, ^3+1), ??-^oo. Therefore by the well known Tauberian theorem, from nan = 0(1) (H, 13+1), n->oo and our other hypothesis Sn->s(A) (%->oo), we have s,,->s(H, (9), n-f co,. that is, Sn-^-s(H, —k), n—>- oo.. Thus our theorem is proved. Next, we will prove the case of a fractional numbsr k for k>l. Theorem 2. If {s»} is summable (A) to s, and if, for a fractional number k, k>l. n'. [n.) J»s-" ° °(1) IJff- -". then {Sn} is summable [H, —k) to s, where k:=l+m, I is a fractional number for 0</<1 and m is a positive integer. Proof. By our assumption. (^) J^_,,,, = 0(1) (ff-, -/), we have. (^) J-S^n = 0(1) (.Hr. -l+l}, that is,. 0 J'"s—= °(1»From the other assumption, {s^} is summable (A) to s ; and so, by the Jakimovski theorem, {s,J is summable (H, —m) to s, or {h\,~l~m')} is summable (H, I) to s, and consequently summable (A) to s too. Thus, from the proposition proved by 0. Szasz in [5]3) {h^~l~)} is also summable (A) to s. Also from our hypothesis,. 0"s"- - °w <ff- -"• and so, by the Jakimovski theorem, {Ji(nl')} is summable [H, —m) to s. Hence, {Sn} is summable (H, —l—m} to s, that is, {s^} is summable {H, —K). Thus our theorem is proved. In conclusion I should like to express my grateful thanks to Dr. G. Sunouchi and Dr. T. Tsuchikura for their kind suggestions.. 3) Theorem of 0. Szasz. If {s,;} is summable (A) to s and {tn} is a regular HausdorfF transform of {s,i} ; then {tn} is summable (A) to s too.. — 10 —.
(5) On the Holder Mean of Negative Order. References. 1. Amnon Jakimovski: Some Tauberian theorems. Pacific Journal 7 (1957), 943-954. 2. C. T. Rajagopal: Simplified proofs of "some Tauberian theorems" of Jakimovski, Pacific Journal 7 (1957), 955-960. 3. L. S. Bosanquet: An extension of a theorem of Andersen, Jour. London Math. Soc. 25 (1950), 7280. 4. A. F. Andersen : Proc. London Math. Soc. (2), 27 (1928), 39-71. 5. 0. Szasz: On the product of two summability methods, Ann. Soc. Pol. JVtath., 25 (1952), 75-84. (1953). 6. G. H. Hardy : Divergent series (Oxford, 1949).. 11.
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