Pacific Journal of Mathematics

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1 Pacific Joural of Mathematics ON A TAUBERIAN THEOREM FOR ABEL SUMMABILITY OTTO SZÁSZ Vol., No. November 95

2 ON A TAUBERIAN THEOREM FOR ABEL SUMMABILITY OTTO SZASZ. Itroductio. I 928 the author proved the followig theorem [2, Sectio 2]: THEOREM A. If p > ad (.) Σ vp l«vl P =0(), -^oo ; the Abel summability of the series Σ = 0 a to s implies its covergece to s. The theorem is the more geeral the smaller p is; it does ot hold for p [2, Sectio ;, pp.9,22]. However, for this case Re'yi proved the followig theorem: THEOREM B. // i lim ~ Σ ^ α v I = / < 00 exists, the Abel summability of Σ = 0 to s. a to s implies covergece of the series 2. Geeralizatio. We give a simpler proof ad at the same time a slight geeralizatio of Theorem B THEOREM. Assume that (2.) V = Σ \a \ v v =0(), v- ad that (2-2) I y.-ί, _><>, m Received April 0, 950. The preparatio of this paper was sposored (i part) by the Office of Naval Research. Pacific J. Math. (95),

3 8 OTTO SZASZ for every sequece m m 9 such that m / > as >. The Abel summa bility to s of Σ Λ =o a implies its covergece to s. Property (2.2) is called slow oscillatio of the sequece V /. Proof of Theorem. We write v=0 v-0 It is easy to verify that, for k 0,,2,, we have (2-3) a_ l - O r +k =-JL-< f r a. k - t r _ l )- ±.-i (k + - v) It is kow [see 2, Sectio 2] that if for a fiite s we have 00 lim Σ a χγl = 5, the (2.) implies σ > s thus, if (2.4) l.u.b. Iσ^.! σ +k I = e, k>o the e» 0. We ow choose (2.5) k =k = [e π /2 ], so that us < πe^ 2 < k + it follows, i view of (2.4), that k + I view of (2.3) our theorem will be proved if we show that k 0, π > oo #

4 ON A TAUBERIAN THEOREM FOR ABEL SUMMABILITY 9 Now k + + o (" + V ) W k + - v., < - {V +k -V B -!), + v ad (2.6) ~ + fe ft - Γ + k } k V + k -\- k π+fe ~ " l usig (2.2) ad (2.5), we see that (2.7) - (V+k -K π _ as π 0 ad π ad thus Theorem is proved. Re'yi observed that the Theorems A ad B are overlappig. We ow show that Theorem icludes ot oly Theorem B,but also Theorem A. Clearly (2.) follows from (.) by Holder's iequality. Furthermore, +k v +k -v = Σ vh < I +k Σ \v=+l l/p = k( P -i)/p 0[( hece, ~V )=-0 (fl - [(Γ 0 as > 0. It ow follows from (2.6) that (2.2) holds; thus (.) implies (2.) ad (2.2), which proves our assertio. A example of a sequece V > 0, ad icreasig, for which (2.2) holds,

5 20 OTTO SZASZ while ~ ι V t oo, is V = log, > 2, because + k \ ) 3. A more geeral result. A geeralizatio of Theorem A is the followig [see 5, p.56] : THEOREM A '. If for some p >, we have (3-D Σ ^P(kl ~O P =0(π), - ^ oo, v-\ the the Abel summability of Σ^= o a implies its covergece to the same value. A aalogue to Theorem is the theorem: THEOREM 2 Assume that (3.2) u = Σ Wkl -α v ) =O(), 2^= αaig? that m (3.3) -Vm~-U >0 as >, > oo. m If ow Σ = 0 a is Abel summable to s, the it coverges to s. Proof of Theorem 2. We have hece [see 5, the Lemma o p. 52] Abel summability of Σ^= o a implies its summability (C,l). From (2.3) we have

6 ON A TAUBERIAN THEOREM FOR ABEL SUMMABILΓΓY 2 from (2.4) ad (2.5) we obtai s -l +k T k T k κ τ χ fe + e /2 Usig the same argumet as i the proof of Theorem, replacig V by U, we fid that (3.4) lim sup s < s. π-oo We ext employ the idetity, similar to (2.3), + TTT Σ ( fe ^ v) α π-v, k = o, l, 2,, ad the iequality a v >. a v ~~ \ a v I The same reasoig as before ow yields (3.5) lim if s > s. Fially (3.4) ad (3.5) prove Theorem 2. It is clear from the proof that coditio (3.3) ca be replaced by as 4. A equivalet result. A glace at the proof of Theorem shows that the followig lemma holds:

7 22 OTTO SZASZ LEMMA l // V is positive ad mootoe icreasig, ad if (4.) V = 0{), as > oo, ad (2.2) holds, the, N m (4.2) -(^-V )»0, as», ^oo. rc We ow prove the iverse: LEMMA 2. // F > 0, ad icreasig, ad if (4.2) holds, the (4.) ad (2.2) ad Proof. We write Kι = ω π, ω > 0, (4.3) ~V ) = ω l ι - ω + ( - - l Let max ω v - the p / ϊp< C0 ltp< CC 9 the F R = O(τι). Suppose ow that p o ; the there are ifiitely may idices m /w^, so that ω m p m for m m^,v,2,3, For these m ad for < m, from (4.3) we get (44) ft K) We ow choose so that Pm

8 ON A TAUBERIAN THEOREM FOR ABEL SUMMABILITY 23 the, usig (4.4), we have " (V m ~V i cotradictio to the assumptio (4.2). It follows that (4.) holds; fially (2.2) follows from (4.), (4.2), ad (4.3). This proves Lemma 2. We ow prove the followig theorem: /2 THEOREM 3. Let U ΣJ = v{ I a v I a v )\ if - ' - ) > - > >oo ad if Σ^= o a is Abel summable 9 the Σ^= o a is coverget to the same value. Proof of Theorem 3. I view of Lemma 2, Theorem 3 icludes Theorem 2; it also icludes Theorem, because of Lemma 2, ad of the iequality Coversely, U m ~U <2(V m ~V ), m>. by Lemma 2, (4.5) implies (3.2) ad (3.3), so that Theorem 3 is equivalet to Theorem 2, ad is thus valid. To show that Theorem is actually more geeral tha Theorem B we give a example of a sequece ω so that ω is icreasig, ω is slowly oscillatig ad ω 0(), but lim ω does ot exist. Let ω Σ v ~ le v i where e v =±; v-\ choose + as log as ω < 3; v, 2,, say. Choose e \ as v u v log as ω ί> 2; v + \ \ 9 * *, ^2> sa y? a( l s o o ^ is clear that ω 0(), ad that lim ω does ot exist. Furthermore, for < ^!, ω Λ t, for Λ ι ^ ^ ~ ^2> ω ^ > a d so o. Now (+l)ω ~ω = (ω ~ ω ) + ω > - -! = - hece ω t. Fially Λ TO - V or m

9 24 OTTO SZASZ hece ω is slowly oscillatig. 5 Aother equivalet result. We first establish the followig lemma. LEMMA 3. Suppose that U > 0 ad icreasig, with UQ 0, ad let (5.) b = ~ (t/ B - t/ -i), >, b 0 = 0 (5.2) B = 2 6 V> π > 0. v = 0 The wheever k A (τz) is so chose that k/ *0, as >0, the two statemets (5.4) B + k -B >0 are equivalet. Proof. From (5.) we have +k Now V-^ x-^ / \ B+k~B = Σ ί>v<- Σ vb v =-(U + k -U ); v=+l v=+l thus (5.3) implies (5.4). Furthermore, B + k B > Γ (tf + fe -f k hece (5.4) implies (5.3). This proves the lemma. We ote that

10 ON A TAUBERIAN THEOREM FOR ABEL SUMMABILITY 25 ad -l U = B ~" X It is a immediate cosequece of Lemma 3 that Theorem 3 is equivalet to the followig theorem (for a direct proof see [4, Theorem IV] ) THEOREM 4. // π + fe \a>v I a v) > 0, as > 0 oo the Abel summabilitγ of Σ w=0 a implies covergece of the series to the same value, A geeralizatio of this theorem to Dirichlet series ad to Laplace itegrals, o differet lies, is give i [3]. REFERENCES. A. Reyi, O a Tauberia theorem of O. Szasz, Acta Uiv. Szeged. Sect. Sci. Math. (946), Szasz, Verallgemeierug eies Littlewoodsche Satzes liber Potezreihe, J. Lodo Math. Soc. 3 (928), _. f Verallgemeierug ud euer Beweis eiίger Satze Tauberscher Art, Sitzugsberichte d. Bayer. Akad. d. Wissechafte zu Muche, 59 (929), , Geeralizatio of two theorems of Hardy ad Littlewood o power series, Duke Math. J., (935), , Itroductio to the theory of diverget series, Uiversity of Ciciati, Ciciati, 944; Hafer, New York, 948. UNIVERSITY OF CINCINNATI AND NATIONAL BUREAU OF STANDARDS, LOS ANGELES

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12 EDITORS HERBERT BUSEMANN R. M. ROBINSON Uiversity of Souther Califoria Uiversity of Califoria Los Ageles 7, Califoria Berkeley 4, Califoria E. F. BECKENBACH, Maagig Editor Uiversity of Califoria Los Ageles 24, Califoria ASSOCIATE EDITORS R. P. DILWORTH P. R. HALMOS BQfRGE JESSEN J. J. STOKER HERBERT FEDERER HEINZ HOPF PAUL LEVY E.G. STRAUS MARSHALL HALL R. D. JAMES GEORGE POLYA KOSAKU YOSIDA SPONSORS UNIVERSITY OF BRITISH COLUMBIA UNIVERSITY OF SOUTHERN CALIFORNIA CALIFORNIA INSTITUTE OF TECHNOLOGY STANFORD UNIVERSITY UNIVERSITY OF CALIFORNIA, BERKELEY WASHINGTON STATE COLLEGE UNIVERSITY OF CALIFORNIA, DAVIS UNIVERSITY OF WASHINGTON UNIVERSITY OF CALIFORNIA, LOS ANGELES * * * UNIVERSITY OF CALIFORNIA, SANTA BARBARA AMERICAN MATHEMATICAL SOCIETY OREGON STATE COLLEGE NATIONAL BUREAU OF STANDARDS, UNIVERSITY OF OREGON INSTITUTE FOR NUMERICAL ANALYSIS Vari-Type Compositio by Cecile Leoard Ruth Stafford With the cooperatio of E. F. Beckebach E. G. Straus Prited i the Uited States of America by Edwards Brothers, Ic., A Arbor, Michiga UNIVERSITY OF CALIFORNIA PRESS BERKELEY AND LOS ANGELES COPYRIGHT 95 BY PACIFIC JOURNAL OF MATHEMATICS

13 Pacific Joural of Mathematics Vol., No. November, 95 Ralph Palmer Agew, Ratio tests for covergece of series Richard Ares ad James Dugudji, Topologies for fuctio spaces B. Arold, Distributive lattices with a third operatio defied R. Big, Cocerig hereditarily idecomposable cotiua David Dekker, Geeralizatios of hypergeodesics A. Dvoretzky, A. Wald ad J. Wolfowitz, Relatios amog certai rages of vector measures Paul Erdős, F. Herzog ad G. Pirai, Schlicht Taylor series whose covergece o the uit circle is uiform but ot absolute Whilhelm Fischer, O Dedekid s fuctio η(τ) Werer Leutert, The heavy sphere supported by a cocetrated force Iva Nive ad H. Zuckerma, O the defiitio of ormal umbers L. Paige, Complete mappigs of fiite groups Otto Szász, O a Tauberia theorem for Abel summability Olga Taussky, Classes of matrices ad quadratic fields F. Tricomi ad A. Erdélyi, The asymptotic expasio of a ratio of gamma fuctios Hassler Whitey, O totally differetiable ad smooth fuctios