On some aspects of summability*

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXV (1985) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXV (1985)

Kanhaiya Prasad (Rae Bareli, India)

On some aspects of summability*

Abstract. The present paper has been devoted to study the (J, ря) summability of some series associated with Fourier series which may be considered as corresponding theorems of Khan [4].

00

1. Let pn > 0 be such that £ p„ diverges, and the radius of convergence

l n= 0

of the power series

00

(!•!) P (z )= £ pnzn

n= 0

be 1. Given any series £ a „ with the sequence of partial sums {s„}, we shall use the notation

(!-2) P,(r)= £ P„snrn

n= 0

and

(L3) J s(r) = Ps{r)/p(r).

If the series on the right of (1.2) is convergent in the right open interval [0, 1), and if

lim J s{r) = S, X~»l _

we say that the series J^a„ or the sequence {s„} is summable (J , pn) to S, where S is finite (Borwein [1], Hardy [2], p. 80).

Particular cases of this method of summability are:

(a) the Abel method: where pn = 1 for all n;

(b) the (Ая) method: where pn is given by

( l - r ) _A_1 = £ pnrn for X > - 1 (|r| < 1);

n= 0

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

(с) the logarithmic method (L): where p„ is given by

00

r ~ 1 l o g ( l - r ) “ 1 = Y pn r".

n — 0

2. Suppose that / is a Lebesgue-integrable function, periodic with period 2л. Let

00

(2.1) / W ~ i « o + Z (я,,cosnx + b„sinnx) n — 1

be its Fourier series.

The series

00

(2.2) Y n (bncosn x — a„sm nx),

n — 1

which is obtained by differentiating (2.1) term by term, is called the first derived series or derived Fourier series of /.

The series

00

(2.3) Y ten sin x — b„cosnx)

n= 1

is called the conjugate series of the Fourier series (2.1).

We shall use the notations:

<p(t) = ? { f ( x + t ) + f ( x - t ) - 2 S } , il/{t) = i { f { x + t ) - f { x - t ) } ,

g ( t ) = f ( x + t ) - f { x - t ) - 2 t f ' { x ) , / ( * ) = - - J

к o+ ita n ^ f

It is known (Zygmund, [8]) that / exists almost everywhere, if / is integrable.

3. Regarding the summability {Ax) of Fourier series and its allied series, various results have been established, among which special mentions may be made of the works of Mishra [6], Khan [5]. The summability (L) of Fourier series and its allied series has been studied by Hsiang [3] and the ( J , pn) summability of Fourier series has been established by Khan [4]. The object of the present paper is to find out some results for the (J , pn) summability of series (2.2) and (2.3). We prove:

Theorem 1. A necessary and sufficient condition fo r the derived series (2.2) o f the Fourier series o f a function f to be summable ( J , pn) to the sum f'(x ), is that

J Im p (relt)dt = o(p(r)), о '

fo r any arbitrary Ô, 0 < Ô < к, as r -> 1 —.

On some aspects o f summability 341

Corollary 1. The (J , pn) summability o f the derived series (2.2) o f the Fourier series o f f at x, is a local property o f /', near x.

Corollary 2. L et the sequence {p„} be positive and decreasing steadily to zero, such that {np„} is bounded. I f

(i)

(ii)

$ \ d g (u )\ = o (tp {l-t)) ( f ^ 0 + ) , 0

*c \dg{u) I . x

j — — = o ( p ( l - t ) ) (t-> 0 + ) t u

fo r any arbitrary ô, 0 < Ô < к, then the derived Fourier series (2.2) o f f is summable ( J , p„) to f ' ( x) at x.

Theorem 2. A necessary and sufficient condition fo r the conjugate series (2.3) o f the Fourier series o f a function f to be summable ( J , pn) to the sum f (x), is that

0 il/(t)

j ~ —RQp(relt) dt = o(p(r)), о 1

fo r any arbitrary ô, 0 < Ô < n, as r -> 1 — .

Corollary 3. T he ( J , pn) summability o f the conjugate series (2.3) o f the Fourier series o f the function f at x, is a local property o f / at x.

Corollary 4. L et the sequence {pn} be positive and decreasing steadily to zero. I f

(i) j \ij/(u)\du = o ( t l+ e p { l - t ) ) (Г-+0 + ), о

where e is a positive number as small as we please, and

(ii) = o ( p ( l - t ) ) ( J - + 0 + ) t и

fo r any arbitrary ô, 0 < Ô < к, then the conjugate series o f the Fourier series o f the function f is summable (J , pn) to f (x) at x.

4. For proving our theorems we will need the following lemmas:

Lemma 1; I f the sequence {p„} is positive and decreasing steadily to zero, then

(i) Ü p„rnsinnt = 0 ( 1 )

n= 0 and

(ii)

n= 0

p„rncosn t = 0(1)

fo r 0 ^ r < 1 and f o r all real t.

P r o o f . Since the series £ r” sinnt and £ r”cosnt are both convergent for 0 ^ r < 1 and for all real t, and the sequence {p„} is positive and decreasing steadily to zero, both parts (i) and (ii) of the lemma follow immediately from Dirichlet’s test of convergence of infinite series.

Lemma 2. I f the sequence {pn} is the same as in Lem m a 1 such that {npH}

is bounded, then

5. P r o o f o f T h e o r e m 1. Let cr„(x) = £ k (b k c o s k x — ak s'mkx) be the nth partial sum of series (2.2). Then following on the lines of Zygmund [8], Prasad [7], we have

= I i + I 2 + o(p(r)), say.

Clearly, 12 = 0(p(r)), by the Riemann-Lebesgue theorem and the regularity of the ( J , pn) method.

fo r 0 ^ r < 1 and f o r all real t.

P r o o f .

lim - У, p„rns in n t= lim Упр„глsin nt

= 0 ( Z ПРпгП) nt

because { np„] is bounded.

П

Thus

Further,

Hence

On some aspects o f summability 343

But

ao OD ac

X

X p.«,wr*-/'w X Pnr"

ao

= Z = ра(г)“ /'(х)р(г).

Therefore

P s (r )-f'(x )p {r ) = - j d^ l m p ( r e it) + o(p(r)) TC O r

from which it is clear that

- x } = / '(* ) if and only if

J - ^ I m p(reil) = o(p(r)).

о *

This completes the proof of Theorem 1.

6. P r o o f o f C o r o l l a r y 1. From the proof of Theorem 1, it is clear

for any arbitrary <5, 0 < Ô < к, as r -> 1 —.

This implies that the ( J , pn) summability of (2.2) depends on the behaviour of f in the nbd of the point x.

This proves Corollary 1.

P r o o f o f C o r o l l a r y 2. We write that

Now

by Lemma 2

and

as t —► 1 —.

Hence the desired result follows from Theorem 1.

7. P r o o f o f T h e o r e m 2. Let П

S0 = 0, Sn = Sn(x) = Y (ams in rn x -b mcosmx), и = 0 , 1 , 2,

m = 1

be the sequence of partial sums of (2.3).

Let us denote by

n — 1

Sg = 0 , S * = S* (x) = Y (a m s*n mx - bm cos mx) + j (a„ sin nx - b„ cos nx),

m = 1

n — 1 , 2 , . . . the modified partial sum of (2.3).

We know (Zygmund [8]) that

~ * 2 ; фи)

S * —f = — f - --- j—cos ntdt n J к о 2 t a n i t

2 (0 „v 2 ^ ( 0

— f ---- - c o s n t-h o ( l) = — f — -cosnt + o (l)

n о t n J0 t

as f --- c o s n t d t-^ 0 , by Riemann-Lebesgue theorem, and thatях о s *

Sn — S * ^ 0 uniformly as n -> oo.

Consequently, (7.1)

Now (7.2)

Y Pn(Sn- S t ) r n = o(p{r)), as r -» 1 = . n= 0

oo 2 00 ô é ( t )

Y

= ~ Z

+

n=0 K n=0 0 f

2 *Ф( 1)

= - f - - ( I p„cosntr")dt + o( X Р„г")

Я 0 ^ n= 0 n= 0

2 î l M I )

= — f ——- R ep(re,')dt-l-o(p(r)).

7Г g t But

Z

X p«(^-^)r"+Z /Us*- 7 V”

n=0 ^{n = 0} n= 0

= o(pM) + I P »(S?- Л г " by (7.1)

n= О

<5

2

= O (p (r)) + - j ^ Re p {ré') dt by (7.2).

71 о ^

On some aspects o f summability 345

Hence the sequence {5„(x)} is summable ( J , pn) to f { x ) if and only if

^ il/ (t)

J — Re p (reй) dt = о (p (r)), о 1

and thus Theorem 2 is proved.

8. P r o o f o f C o r o l l a r y 3. From the proof of Theorem 2, it is evident that

® ~ 2 ô \l/(t)

Ps (r) = Z P n ( S n - I ) r n = - f Re p { r é 1)+ o (p (r)),

n= о я 6 t

for any arbitrary <5, 0 < ô < n, as r -» 1 —.

This implies that the (J , p„) summability of the conjugate series (2.3) depends upon the behaviour of the conjugate function / in the nbd of the point x and this proves Corollary 3.

P r o o f o f C o r o l l a r y 4. We write

Ô è ( t ) 1 -r à

J — - R e p ( r e lt) d t = { + { = P 1{r) + P 2{r), say.

0 * 0 1 — r

Now

1~r M (t)I

P i(r) = 0 ( l ) f --- —dt by Lemma 1 (ii) n t

=

0

1

|o(r1+sp(l-t))l + j i-o(t1+£p(l-t))<fr

t J О О r

= 0 ( l ) [ o ( l ) + o ( l ) ] = o ( l ) . By Lemma 1 (ii), we have

P2(r) = 0(l)^J ^j^-dtS j = o(p{r))t

Hence Corollary 4 follows from Theorem 2.

A ck n o w le d g e m e n ts. The author is grateful to late Dr. S. R. Sinha, Department of Mathematics, University of Allahabad for his kind encouragement during the preparation of this paper. The author is also very much thankful to the learned referee whose valuable comments and suggestions improved the presentation of the paper.

References

[1] D. B o rw ein , On methods o f summability based on power series, Proc. Roy. Soc. Edinburgh 66 (1957), 342-348.

[2] G. H. H ardy, Divergent Series, Oxford 1949.

[3] F. C. H siang, Summability (L) o f the Fourier series, Bull. Amer. Math. Soc. 67 (1961), 150- 153.

[4] F. M. K h an , On (J , p„) summability o f Fourier series, Proc. Edinburgh Math. Soc. 18 (Series II), Part I (1972), 13-17.

[5] H. H. K h an , On some Aspects o f Summability, Indian J. Pure Appl. Math. 6 No. 12 (1975), 1468-1472.

[6] В. P. M ish ra , The Summability (Ax) o f the Fourier series and the first differentiated series, Math. Student 11 A (1972), 331-337.

[7] K. P rasad , On the (N, p£) summability o f Fourier series and its allied series, Indian J. Pure Appl. Math. 9 (1) (1978), 47-60.

[8] A. Zygm und, Trigonometric series, Vol. II, Cambridge 1959.

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