174. A Tauberian Theorem for (J,,fin) Summability*)

Transcription

1 No, 10] A Tauberia Theore for (J,,fi) Suability*) By Kazuo IsHIGURo Departet of Matheatics, Hokkaido Uiversity, Sapporo (Co. by Kijiro KUNUGI, M.J,A., Dec. 12, 1964) 1. We suppose throughout ad the radius of covergece p? 0, p= oo, of the power series is 1. Give ay series p(x) = ~, px =U (1) 1Ja~ with the sequece of partial sus {s}, we shall use the otatio: (2) p3(x)= psx If the series (2) is coverget i the ope iterval (0, 1), ad if ur p(x)3 x--4-0 p(x) we say the series ~, a or the sequece {s} is sable (J, p) to s. As is well kow, this ethod of suability is regular. (See, Borwei [1], Hardy [2], p. 80.) Now we write ad P = po " p1.~'... ±p,,, 1,..., (3) t=-- P 1 /=o pvsv,, 1, with p >0. If {t} is coverget to s, we say the series E a or the sequece {sj is suable (N, p) to s. This ethod of suability is also regular, ad is equivalet to the Riesz ethod (1?, P-1, 1) (See, Hardy [2], pp. 57, 86, Jurkat [4], Kutter [5,6].) We shall first state the followig Theore 1. (N, p) iplies'' (J, p). *) Dedicated to Professor Kijiro Kuugi for his 60thBirthday,. 1) Give two suability ethods A, B, we say A iplies B if ay series or sequece suable A is suable B to the sae su. We say A is equivalet to B if A iplies B ad B iplies A.

2 .808 K. IsHIGuRo [Vol. 40, The proof of this theore ay be deduced fro a geeral theore, however we shall give here a sketch of a brief proof. (See, e.g., Hobso [3], p. 181.) Fro (3) we get with t_1= P_1= 0. tfp - t9-1p_1- lros = 0, 1,..., Hece ps(x) _ psx _ (t,p-t-1-1)x =(1-x) tpx fro the assuptio of the theore. Now sice ps(x) _ (1-x) p(x) _ p x tpx _, 00 tpx P x.- Pt(x) P(x) we have, agai fro the assuptio of the theore, li ps(x) = li Pt(x) = s, p(x) x--*1- P(x) which proves the theore. 2. Cocerig the (N, p) suability we kow the followig Tauberia Theore 2. Suppose p,>o, - 0, 1,..., a - ad the series (1) is suable (R, P, 1). The (1) coverges to the sae su. (See, Hardy [2], p.124.) Sice (N, p) iplies (J, p), we ca expect Tauberia theores of the siilar type for the (J, p) suability. We shall prove here the followig Theore 3. Suppose tp (4) (1) for -->oo, p 1- p

3 f No, 10] Tauberia Theore for (J, p) Suability 809 (5) 0<p<M, -0,1,..., with soe costat M, ad (6) P = 0(1). Suppose the series (1) is suable (J, p) to s, ad (7) a - o P Proof. We have, for 0<x<1, The (1) coverges to s. s p (x) - 0 SPx - spx p(x) ~ p x ~ Px _ (s - s)px lrox Here we get -1 (S- s)px (S- s)px = + =+1 Px ~ Px =I+J, say. px <_ 1 P1 a1 I p0 + p2 I a2 I (p0 + P1) ~ P px 1 + P I a I (p0+ P P-1) p P2 ad therefore, whe x =1- - ~ P 1 ~ P p1 + p2 I a2 I P p I a P-1 P2 p Now, fro (7), we see

4 810 K. ISHIGURO [Vol. 40, a P -1 = =o(1) for -~co. pi Hece, accordig to (4), we have (8) I=o(1) for --oo. Next we shall estiate J. For ay ~, >O, let be so chose a E Pp for >, the Therefore s - ~ ~ ~ '+1 p+1.~ p =Q, ~say. we have + p P Qpx (9) + J! =~ 3. px =o Qp 1-1 =+1, - o, ( i_i) g =o if x be chose to be equal to 1-1. Sice we get Q < P P P - P._ 1 P ~ 1 ~ Pp 1- ~ J I ~ P =-f-1 ~ 1 p ~Pi- P 1 =+1 Pp 1- ~ 1 p 1 1 ~ P P =+1 1-1\2) fro (4) ad (5). Also, agai usig (5), we have 2) We use M to deote a costat, possibly differet at each occurrece.

5 No. 10] Tauberia Theore for (J, p) Suability 811 J<~M1 (- 2 ~I1_1 P = < EM-1-- ~x 1-1 xdx P2 <~M 2 P 2 for large, fro (6). Lettig icrease idefiitely, we have h s = li p8(x) = s p(x) fro (8) ad (10), which proves the theore. 3. The assuptios of Theore 3 see to be very coplicated, however it follows fro this theore the followig Corollary. Suppose there exist two ubers a, M such (11) 0<ap<M,, 1, Suppose the series (1) is suable (J, p) to s, ad (7) a=~ p P The (1) coverges to s. Proof. It suffices to prove (11) iplies (4) ad (6). Fro (11), we see ~ 0 p?ti <M ~ o~ 1-0 ~p 1--` ~p 1-- ~ <_ M ^o 0o 1 1 < M -< a for large. Fially we see, fro (11), < <1 P (+1)a a for large. We reach the desired coclusio. Reark. I the corollary, the coditio (7) ay be replaced by

6 812 K. IsHIGURO [Vol. 40, a o(1-). Refereces [1] [2] [3] [4] [5] [6] D. Borwei: O ethods of suability based o power series. Proc. Roy. Soc. Ediburgh, Sect. A, 64, (1957). G. H. Hardy: Diverget Series. Oxford (1949). E. W. Hobso: The Theory of Fuctios of a Real Variable ad the Theory of Fourier Series, vol. 2. Cabridge (1926). W. Jurkat: Uber Rieszsche Mittel it ustetige Paraeter. Math. Z., 55, 8-12 (1952). B. Kutter: A Tauberia theore for discotiuous Riesz eas (I). Jour. Lodo Math. Soc., 38, (1963). A Tauberia theore for discotiuous Riesz eas (II). Ibid., 39, (1964).