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OF THE
INDIAN ACADEMY OF MATHEMATICS
Volume 16
1994
1994
No. 2
THE JOURNAL OF THE INDIAN ACADEMY OF MATHEMATICS (A half-yearly Research Journal) Editor
R. K. SAXENA Secretary : C. L. PARIHAR Editorial Board R. N. JAIN D. V. JAISWAL V. M. BIIISE N. V. DESHPANDE
S. V. MORE V. D. Jim C. K. DAVE
M. K. DUBE? K. B. RAJOLE S. C. BHATNAGAR K. SHANTARAM
Advisory Committee
R. G. BUSCHMAN K. NISHIMOTO
(U.S.A.) (JAPAN)
H. P. DIXIT
(NEW DELHI)
It. P. AGRAWAL
(LUCKNOW)
N. B. YENGIRARIAN
(RUSSIA)
K. N. SRIVASTAVA
HAROLD EXTON
(U.K.)
J.N. KAPUR
(NEW DELHI)
(BHOPAL)
H. M. SRIVASTAVA
(CANADA)
D. K. SINHA
(JADA VP UR)
KIYOSHI ISEKI
(JAPAN)
GOPI AHUJA
(LUCKNOW)
V. P. SAXENA
(GWALIOR)
K. M. SAKSENA
(KANPUR)
Papers are accepted from members of the Academy in good standing. In case of joint authorship each author should be a member of the Academy. Papers intended for publication in the journal should be sent in duplicate to the Secretary, 15, Kaushaliyapuri, Chitawad Road Indore, 452001. They must be type-written, double spaced, with references listed in the alphabetical order in the end. Normally papers, with more than twelve typed pages, are not considered. Authors are advised to retain a copy of anything they may send for publication. To meet the partial cost of publication they will be charged a page-charge of Rs. 75/($ 15.00) per page, payable in advance, after acceptance of article. Authors (the first in the case of joint paper) are entitled to 25 reprints free of charge. Price of Complete Volume of the journal is Rs. 120/- in India and U.S. $ 60 outside India. Single copy (current) is priced Rs. 60/- in India and U.S. $ 30 outside India. Back volumes are priced at Rs. 75/- (U. S. $ 38) per number. The subscription payable in advance should be sent to the Secretary, 15, Kaushaliyapuri, Chitawad Road, 'adore, 452001 by Bank-Draft only in the name of "Indian Academy of Mathematics, 'adore" and Redrawee Bank should beState Bank of India or its subsidiaries or any nationalised Bank situated at Iadore. Postal orders are not accepted.
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Vol. 16, No. 2 (1994)
Preen Chandra and Miss Varsha Kshirsagar
ABSOLUTE NoRLUND SUMMABILITY OF FOURIER-JACOBI SERIES AT FRONTIER POINT
In this paper, the authors obtain a general result for absolute Norlund summability of Fourier-Jacobi series at the frontier point under the local conditions which not only extend some known results (see [2] and [3]) but also yields some new results. 1. Definitions and Notations. Let B an be any given infinite series with the n=0
sequence of its partial sums (s„) and let (q„) b: a sequence of positive real numbers such that
Then the series E an is said to be absolutely summable (N, g„) or summable n -o
- -(11Q.) IN, gn i if the series E Itn —t,,.. 1 1 is convergent, where to , n-0
sk and by
convention 1_,=0. Let f(x) be a function defined Li the closed interval [-1, 1] such that the integral
If(x)1 dx, (a> —1,
p> -
J1 1
is convergent. The Fourier-Jacobi expansion corresponding to f(x) is given by (see Szegt5 [4], pp. 243)
f(x) ,
E
4-0
as pn(a ' 13) (x),
(1.1)
wnere a> — t, )f>
-
1,
ice /3) (t) dt, (1—t)a (1+ t)/3 f (t) pn ' ,V((n 4- -I- 1))
200+1
/((n
(3 -I-1))
gn 2n+ a+ p+1. 1/((n+1)) V((n+a+13+1))
(a, /3 ) and p n (x) are Jacobi polynomials of order a, 11 and are defined by 2a+P (1 —2 vt+ / 2) -1 /2 [1 — t+.0(1 —2xt + 12 ))ra -14+ VW —2xt OM -13 (a, (a) (x)
t
p„
n
-
We use the following rotations : F(ch)---(f(cos
— A) (sin (4)/2)1 7 a-" (cos (0/2)) 2P+1,
(1 2)
where A is some fixed number, FAO= ; IF(91.)j 0 (a, P) nPm
(1.3) (a, P')
(1) Pm
Sn (1, cos 0)—
(cos (k)
(1.4)
gm
2. Introductions. In 1971 Gupta [2] studied the localization problem for the summability IN, p,1 1 at the frontier point x=1 of the linear interval [-1, 1] for Jacobi series (1.1). A more generalized version of his result, in silghtly modified form given by Singh [3], is : Theorem A.
If (q.) is a positive sequence such that the sequence (9n-l-t
)c by
(2.1)
< CO .
(2.2)
and the series, n 118+0,
Then IN, q„I summability of the series (1.1) depends on the behaviour of the funcprovided a> 'ion f in the immediate neighbourhood of the point $> and the antipole condition
C
_F.:tit/2/3-10i -1
dx -2 . Thus Theorem B, as such, gives wrong information for µ-{-a=2 as remarked above. On the otherhand, if we suppose
3,
' should be `IL +a> 2 : thc:n (2.4) of Theorem B becomes re-
dundant and does not provide any answer for µ+a‹3. These questions are answered by the theorem which we propose to prove. In fact we obtained a result for 1N. , q„I summability of the Jacobi series (1.1) at a point x=1 under the local
condition imposed upon the generating function of the Jacobi series which, as particular cases, yields far reaching results as corollaries. Precisely, we prove the following : Theorem.
1
Let
p> - and let g and H be positive monotonic increas-
ing and monotonic decreasing functions respectively such that as (-4.0+ Fi (t)=0(g(t)),
(2.6)
wad for 00)
in (2.6). Now take H(t)=0(1) for all t and Tn —na+" for large n.
Then we
observe that (2.7) and (2.9) are satisfied and (2.8) reduces to (2.2) for Hence we get the corollary.
Kemargs. tills may be observed that the above corollary exhibits that IN, g n i summability of (1.1) depends upon the behaviour of the generating function f in the immodiate neighbourhood of the point x=1 whenever (2.1) through (2.3) hold for 1 Therefore this corollary may be compared with Theorem A which holds 2 for a> —1. 2 We further remark that this can not be deduced from Theorem B as one can see that the conditions (2.5) and (5.1) are non-comparable. Therefore Corollary 1
yields a new result for IN, g„I summability. 1
Corollary 2. Let
/3> — and let g(t) be positive monotonic increasing
such that as t--)-0+ F1 (t)-0ttp+2 0.+1 1
If a+a
Go_ 2a + 1 Or.
(5.2)
3 and positive sequence (q.) satisfies (2.1) and the following : 1 (5.3)
E (n`S 1Q„)-- and 2—•p if 'L. 1 a< then the Jacobi series (1.1) at 2 x--1 is IN gn I surnmaO le whenever (2.3) holds. - -
-
2'
Proof.
Assume in (2.6)
g (0= tit+ 2 a+1
(L-F2a+1;-->-0).
Then we may take in (2.7)
r0 H(t) =-
, 1 ,
I
IF-Fa-3/2
3 p+a>2 IL
_Fa .j
Using these values, we observe that (2.9) holds with T,,=n8 , where 8—max 1 1 However if g + a>. ' then a+2>2—µ [2-p, >2—ii and hence 8=a+-2. In 2 a+2 3
-
3
*In the caseA-1-a>— this restriction is not required. 2'
case /z+a
