r.(x) = L î P.-.U.W (P„# ). On the uniform Nôrlund summability of Legendre series*

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXV (1985)

Ka n h a iy a Pr a s a d and Su r e s h Ch a n d r a Lo d h i Ra jp u t (Allahabad, India)

On the uniform Nôrlund summability of Legendre series*

Abstract. Saxena [6 ] and Dwivedi [1] studied the uniform harmonie summability of Fourier series and uniform harmonic summability of Legendre series respectively. In the present paper the authors have studied the uniform Nôrlund summability of Legendre series which is a generalization of Dwivedi’s result [1].

1.1. Let

u0 (x) + u1 (x) + i i2 (x) + . . .

be an infinite series, and

t7v(x) = M o M + Hj (x) + . . . + mv(x).

Let {Pn} be a sequence of constants, real or complex and let Pn = Po + Pi+P2+ ••• +Pn>

We define sequence-to-sequence transformation

r.(x) = L î P.-.U.W (P„# ⁰).

* n v= 0 If r„(x) -+ U(x) as n -*• oo, we write

£ M * ) = U(x) (N , Pn),

v — 0

or

Uv(x ) U (x) (N, pn).

If Lim [fn(x)—I/(x)] = 0 uniformly in a set E, then we say that the я 00

series un(x) is summable (N , pn) uniformly in E to the sum U(x).

This paper is dedicated to Professor W. Orlicz on the occasion of his 80th birthday.

116 К. Prasad and S. Ch. L. Rajput

1.2. The Legendre series, associated with a Lebesgue integrable function in the interval defined by — 1 ^ x < 1, is

00

(1.2.1) f ( x ) ~ I a.P.(x),

n= 0 where

an = { *+i ) J f ( x ) P n{x)dx

and the и-th Legendre polynomial P„(x) is defined by the following expansion

( 1 - 2xz + z 2Ÿ '2 = P M )Z ''•

We use the following notations:

Ф(0 = ф(в, t) = / {cos( 0 -г )} - / ( c o s в), W(t) = j №(u)\du, t = [1 /0 , 0

where [t] denotes the integral part of t.

13. In [6] Saxena, extablished the following theorem for the uniforn harmonic summability of a Fourier series.

Th e o r e m A. I f

j * 1 / ( x - I -u) + / ( x — u) — 2s\ du = о ^ — ^{^} as 0

uniformly in a set E in which s = s (x) is bounded, then the Fourier series of a function f (t), is summable by harmonic means uniformly in E to the sum s.

In [1] Dwivedi gave a corresponding result for uniform harmonic summability of Legendre series.

Namely, he proved the following

Th e o r e m B. I f

j l Г ( х ± и ) Ч Ш » = о { ~ 1 0

as t -* 0 +

uniformly in a set E defined in the interval ( — 1, +1), in which f (x) is bounded, as f - + 0 + , then series (1.2.1) is summable by harmonic means uniformly in E to the sum f (x).

The object of the present paper is to extend the above result to the case

of uniform Norlund summability of Legendre series in the form of the following theorem:

Theorem. I f

(1-3.1) I f ( x ± u ) - f { x ) \ d u = о 0

X{l/t)t

Pr as t -> 0 + ,

uniformly in a set E defined in the interval ( — 1, +1) in which f (x) is bounded, then series (1.2.1) is summable (N, Pn) uniformly in E to the sum /(x ).

Here A(r) is a positive, monotonie, non-decreasing function and {P„} is a real, non-negative monotonie non-increasing sequence such that P„ -* oo as n -* oo and

X(n) log n = о (Pn) as n —> oo.

1.4. Following lemmas are required for the proof of our theorem.

Lemma 1 ([5]).

(1.4.1) £ (2v + l)P ,(x )P ,(y ) = ( n + l)

v = 0

Pn+1(y)Pn(x)-Pn(y)Pn+i(x) У-

This identity is known as Christoffel’s formula of summation.

Lemma 2. Under condition (1.3.1), we have t

(1.4.2) j* I / (cos (9 — v)} —/ (cos 0)| dv = о о

A(1 ft)t

~pT as t —► 0 -J-, where x = cos 0, x + u = cos <p, and 9 — (p = v.

The proof of this lemma follows on the line of Foa [2]. * Lemma 3. For 0 ^ a < b ^ oo, 0 < t ^ n and any n,

(1.4.3) I J] Pk Qxp(i(n-k)t)\ ^ APt,

k = a

where A is an absolute constant (McFadden [4]).

15. P r o o f of th e th e o re m . The nth partial sum of the series (1.2.1) is + 1

C , , £ n , , 1^{, , 1 4 I „} , P n + l b ) P n ( x ) - P M P n +1 ( x ) J

Sn( x) = L av Pv{x) = i ( n + l ) f { y ) --- dy

v = 0 J ^{У - Х}

- 1

by Lemma 1.

118 К. Prasad and S. Ch. L. Rajput

Putting f (y) = 1, it can be easily seen that +1

, 1, . 1, I Pn+i( y)Pn(x)-P„(y)Pn+Ax) J l = i ( n + l ) --- --- dy.

y - x Therefore

- 1

+1

S „ (x )-/(x ) = i ( n + l ) I U W - f ( x ) l - 1

and so

+1

Pn + 1 (y) Pn (x) - P„ Q>) Pn + 1 (*)

У - х dy

Sn- k( x ) - f ( x ) = i ( n - k + 1) - 1

Cf(y)-f(x)~] X

P n - k + l ( y ) P n - k ( x ) - P H- k( y)Pn-k+l ( x)

y — x dy.

Let us take a positive number less than 1 and consider it as the sum of two other positive numbers ц and <5. Let d be another positive number such that 0 < d < и and let цх and ц'х be two continuous functions of x within ( —1, 4-1) which lie within the limits

d ^ fxx ^ pi, d ^ /4 ^ ц.

Therefore, for — 1 + S ^ x ^ 1 — 5, we have

x - n x x + n'x + 1

(1.5.1) S , , - * ( x ) - / ( x ) = i ( i i - * + l)

+ J +

- 1 х - ц х Х + Ц х

Pn-k+i(y)Pn- k{ x )- Pn-k{y)Pn-k+i(x) x ---

y — x

= Ап-к(х) + Вп-к(х) + Сп..к(х), say.

dy

Hobson has shown that uniformly in — l + S ^ x ^ l — S (1.5.2) Lim Л„_к(х) = 0 and Lim C„_k(x) = 0.

n->oo n-*oo

Now we suppose x = cos в, у = cos Ф, 0 ^ 9 < n, 0 < Ф < я, 1—5

= cos q, 1 — (m4-<5) = 1 — S = cos(f? + 5), 0 < q < j u , 0 < a; g + a < i n . Thus if rj denotes the minimum of

[arc cos и — arc (и + /л)]

for u in ( — 1, 1 — u), we have on the lines of Sansone e+q

B„_k(cos 0) = i ( n - / c + l ) j* /(cos $ )-/(cO S 0)x o-t,

P„_k+1(cos #)P„_*(cos 0 ) - P „ _ k(cos # )P „_ k+1(cos 0) . x --- --- --- sin Ф аФ

cos Ф — cos 0 in which Q + a ^ 0 < я —(g + a); 0 < rj ^ a.

With successive transformation, we get (1.5.3)

where

D n - M =

B„_k(cos 0) = D„_k(0) + £„_k(0), say,

e+^

1 / (cos Ф) —/ (cos 0) 27usin1/20 J sin1/2(0 — Ф)

е-ч

sin (n—к +1) (0 — Ф) sin1/2 Ф йФ

and obviously £„_k(0) = 0(1) as n- * oo uniformly, where x lies within ( —1 + S , 1— S), i.e., in the set E.

Putting (0 — Ф) = t, we get

(1.5.4) Д ,_к(0) = 1 я sin1/20

/ {cos (0 — t)} —/ (cos 0) sin \ t

xsin(n — k + l)f sin 1,2(0 — t)dt.

Thus, we have from (1.5.1) to (1.5.4)

S ._ ,< * ) - / ( * ) ---A ^ : ^ 1 С^ --/> Ь ^ С° 5Й )Х

я sin1/20 sm ^ t

xsin(n — k + l ) t sin1/2(0 — t)dt + o(l).

Now

T Î f * { S . - , W - / ( x ) }

r n k = 0

1 " 1 Л , k? o P* Я sin1/2 0 X

■J

120 К. Prasad and S. Ch. L. Rajput

uniformly in E

—^TJTb

к sinl/z в P„

" s i n ( n ~ k + l ) t

X I p k — ^ r r .— d t + o W

k = О Ь Ш 2 1

uniformly in E

= O [ J № ( /) ||A U 0 |A ] + o(1) 0

uniformly in E

i h ч

= o [ j \ ф т м м ^ ] + о [ $ \ ф т м м ^ ] + о ( 1 )

О 1 In

uniformly in E

= 0 ( I l ) + 0 { I 2) + o(l), say, where

1 " sin(n — k + l)t

" . W - p I sin- r >- L

r „ k = 0 Sin 2 I

In order to prove the theorem we have to show that under assumptions of the theorem,

(1.5.5) / = 0 ( 1 ) and / 2 = o ( l ) as n-> oo uniformly in E.

Now, uniformly in 0 ^ t ^ 1/n,

So

(1.5.6)

N„(t) = O(n).

1 In

h = 1^(011^(01 dt = 0 i/«

n j \ij/(t)\dt 0

= 0 À(ri) nP„

= o(l), as n->oo

uniformly in E since nP„ ^ P„. Now, uniformly in 0 < t < n, we have N„(t) = 0 [ P z/ t P J .

So

" 1 - _i

0

I . \w(t)\ — dt i

1 In l / n

Г l 1 pJ " г l Г р т 1

0 k h - r u ⁺ 0 [p.

1 In

+ 0

+ o 1 t PT

1 Л J P, t2

1 In

+

‘ I f 1

T ._l/n

J

+

4 -0 l ГЯ(1/0г l

p p

1 n J л T 1 In

- \dPTI

= o (l) + 0

o(l)4-o

Ц Щ dt + o

L Pn J

A (i/o

№

l/n l/n

ft

1 , f î " l f

— Цп) - dt 4 -0

I J t Pn ..

Ц Щ m

l/n l/n

= o(l) + o

= o(l)4-o

— Я (и) {log t}\ln

■* и

4 - 0 \dPt I

n J l/n

'А(Л)

(log // + log и)

~ 1 Д + 0

- p I p*

_ r n k= 0

= o(l) + o (l) + o(l) = o(l) as n -> оo, uniformly in £.

Hence from (1.5.6) and (1.5.7) we get (1.5.5) and this completes the proof of the theorem.

R e m a rk . If we take À(t) as a unit function and P„ = l/( n + l) , then our theorem reduces to Theorem B.

Acknowledgment. The authors are indebted to the referee for his useful suggestions.

122 К. Prasad and S. Ch. L. Rajput

References

[1] G. K. D w iv e d i, On the uniform harmonic summability of Legendre series, Ph. D. Theses, Banaras Hindu University (1970).

[2] A. F о a, Sulla Sommabilita forte della serie di Legendre, Boll. Univ. Math. Ital. (2), 5 (1943), 18-27.

[3] E. W. H o b s o n , On the representation of a function a series o f Legendre's functions, Proc.

bond. Math. Soc. (2) 7 (1909), 24-29.

[4] L. M c F a d d e n , Absolute Norlund summability, Duke Math. J. 9 (1942), 168-207.

[5 ] G. S a n s o n e , Orthogonal functions, English Edition, 179 (1959), 227-233.

[6 ] A. S a x e n a , On uniform harmonic summability o f Fourier series and its conjugate series, Proc. Nat. Inst. Sci. India 31 (1965), 303-310.

F. G. COLLEGE RAE BARELI 229001, INDIA and

DEPARTMENT O F MATHEMATICS

V. S. MEHTA COLLEGE OF SCIENCE, BHARWARI ALLAHABAD (U P.) INDIA