b. c. Singhai - on the Cesaro Summablity of He Ultraspherical Series

7/30/2019 b. c. Singhai - on the Cesaro Summablity of He Ultraspherical Series

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BOLLETTINO

UNIONE MATEMATICA ITALIANA

B. C. Singhai

On the Cesro summability of the

ultraspherical series.

Bollettino dellUnione Matematica Italiana, Serie 3, Vol. 16(1961), n.3, p. 207217.

Zanichelli

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Bollettino dellUnione Matematica Italiana, Zanichelli, 1961.


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SEZIONE SCIENTIFICABREVI NOTE

On the Cesro summability of the ultraspherical series.By B. C. SIJSTTEAI (Sagar, India) (*)

Summary* - Theauthor hasobtained theoremsfor Cesro summability ofthe ultraspherical series which extend and gnralise the results ofwang [6 and 7] ofFourier series.1. Let f(6, cp) be a function defined for the range 0 < 6


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2 0 8 B. C. SNGHAIa n d

0.The absolute integrability of the integrand in (1.2) is assumedthroughout.The author has obtained theorems for CSSRO summability ofthe series (1.1) analogons to those of I Z U M I and ScnsTouCHi [1]. Theobject of this paper is to extend and generalize the results of

WAISTG [6 and 7] for the same series.We prove the following:THEOREM 1 - If v > p > 0 and

for 2X" > 1 and 0 < X< 1.T h e n the s e r i e s (1.1) is s u m m a b l e (c, oc -+- X) at the p o i n t (0,to the sum A, ^wherev(m 1) -+- p

m pand w is a positive integer such that m>$ > m 1.

THEOREM 2. - If p > 1 andJ2X

for B 1 < a < p and 0 < X < 1T h e n the s e r i e s (1.1) is s u m m a b l e (c, oc -+-X) at the p o i n t (9,0)to the sum A.2 . We r e q u i r e the f o l l o w i n g L e m m a s :L E M M A 1. - In o r d e r t h a t the s e r i e s (1.1) be s u m m a b l e (c, k) tat h e sum Ay it is s u f f i ci ent t ha t the i n t g r a l

o = 0(1)0

fo r 0 < o < u and for ea ch h> X.L E M M A 2. - Let Sn(*>>) d n o t e the n C E S R O raean ofjrorder|fc oft h e s e r i e s


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ON THE CESBO SXJMMABILITY OF THE ULTRASPHERICAL SERIES 2 09

T h e n we have , for X > 0 and p>0,0{n 2X+P+1) for 0 < o> < 7T, h> 0,

= =

X -f- 17cand

L E M M A 3. - For a non-integral5 m + cr (0 < a < 1),

w e h a v eA

= m + i ( A W n W ) ( A ) / JL E M M A 4. -

, u) =w h e r e

L e m m a s 1, 2, 3, are due to O B R E C H K O F F [3]and L e m m a 4 isk n o w n [5].3 . P r o o f of T h e o r e m 1.I n o r d e r to p r o v e the t h e o r e m , it is s u f f i ci ent to s h o w t h a t

o

l im ! v(tt)sV

u n d e r the c o n d i t i o n s of the t h e o r e m .


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2 1 0 B. C . SINGHAIWe have the following inequality:

S > a > m 1 [ F. T . W A N G ]

Also we haveT h e n

5f

0

(3.2)

= ! + (1)J.since a >- -m- 1,(3.1) I = o(l) as M- o o .

We write

o

n S= J " + - J =0 1

5! - < D - ^ -


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ON T H E C E S RO ST J M M A B T L I T Y O F T H E U L T R A S P H E R I C A L S E R I E S

0= o (1) as n ^ oo .

Al s o f we W r i t e

T h e n

't)

= o(l) as n ^ o o .W h e n p i s n o t a n i n t e g e r

' . = = 1 , u)du.


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2 12 B. C. SINGHAIwhere

n> u) = f(^ ZT pj (' - ^ "N o w

(3.4)' = [

no /

= o(l) asW he n p = -m is an integ er

in(3.5) J, = O

= = o [ i v i w l= o ( l ) as n o o .

C o m b in in g (3.1), (3.2) , (3.3), (3.4) an d (3.5) th e re su lt is pr ov ed .4. Proof of Theorem 2.If we put

and suppose for a non-integral og - w ^ ff (o < ff < 1),


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ON THE CESRO SUllMABlLTTY OF THE TJI/TKASPHERICAL SERIES 2 1 3

Also let us first take the case when m > 1, thenA

% / ^

sice a>> Sand S>>m, hence m.Thus(4.1) I =: o(l) as n -* oo.

ow

J

= ( J m ^ i ( A ) s w (A) /o

But(4.21

NowJ " g = since a > m.

A r A=j s(u)s(n\u)du = + =Ia

[by Lemma 3]


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2 1 4 B. C . SINGHAIw h e r e

a o a ir =' S - H 2X - H 1 " " p -i -2XB a t w e have [1].

t( 4 . 3 ) c p g ( f ) = - ^ i _ _ ) < p ( M ) c w = 0

O

I f w e p u tB = 1 -*- o

t hen ( 4.3) i s equ i va l ent t oJ+2X(4.4)

so OS(te) is integrable in the sense of C A T I C H Y - L E B E S G U E . T h u s b y(4.4), w e g e t :f f V+2X e 1 1\ / J (4.5) /


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ON THE CESEO SUMMABILITY OF THE TJLTRASPHERICAL SERIES 2 1 5

= o(l) as %-* oo.

(4.7) I, = GafttJsSfV)** = + = o

ky= o\u a~P+ 1 O(fl2^++i\ \nL lo= o(1) as n -+oo.

( 4 . 8 ) k,=o\ fw*-

= o(l) as w-> oo.

(4.9) w ==


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21 6 B . C. S I J N T G H A I

\ - \ *(

Te'

=r o(l) as ^ -* oo.

(4.10) du

= o (n 5-*

na~ p= o ( l ) a s n > oo.

C o m b in in g (4.1), (4.2 ), (4.3), (4.4), (4.5), (4.6), (4.7), (4.8), (4.9), and(4-10) the r e s u l t is p r o v e d .W h e n o is an i n t e g e r , say or^m.(4.11)*= 2(-l)P-i!P=i

= 0(1).


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ON THE CESARO SUMMABILITY OF THE ULTRASPHERICAL SERIES 2l7

When m = 0A

(4.12) / Q)(i(u)s{n\ii)du= 0>1(A)sLo)(A) - /o o"

= o(l) as before.When 6 = 1 i.e. o = O

andt

oBy the help of (4.1), (4.10), we have(4.13) * = o(l)

I am much indebted to Prof. M. L. M I S B A for his kind interestin the prparation of this paper.

KEFEEJEJSTCES[1] IZUMIJ S. & SUNOUCHI, Gr., Notes on Fourier Analysis (XXXIX);

Theorems concerning Csaro sumrnability, Tohoku Math. Jour. , Vol.1-2, second series 1949-51, (313-326).[2] KOGBETLIANTZ, E., Recherches sur la sommabilit dessries ultrasph-nque par la methode des moyennes arithmtiques, Jour Math., (9),3, 1924, (107-187).[3] OBRECHKOFE, Sur la sommation de la sries ultrasphrique par lamethode desmoyennes arithmtiques, Hendiconti del Circolo Materna-tico di Palermo., 59, 1932. (266-287).[4] SANSONE, Gr. Orthogonal Functions, Revised English dition (1959)[5] SINGHAI, B. C. Csaro Summability of ultraspherical series* to appearshortly in Annali di Matematica pura ed applicata., (1961).[6] WANG, F. T. A note on Csaro summability of Fourier series, Ann.Math., 44, 1943(397-400).[7] WANG, F. T. A remark ou (C) summability of {Fourier series, Jour.London Math. Soc, 22, 1947,(40-47).

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b. c. Singhai - on the Cesaro Summablity of He Ultraspherical Series