On the Nôrlund summability of orthogonal series

ROCZNIK1 POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVI (1986)

S. R . Ag r a w a l* a n d Pr a g n a S . Ka n t a w a l a* * ( B a r o d a )

On the Nôrlund summability of orthogonal series

L e t U o + U j + . . . + U „ + . . . b e a g iv e n s e r ie s w it h p a r tia l s u m s s n a n d le t { p n} b e a s e q u e n c e o f n o n - n e g a t iv e r e a l n u m b e r s .

00

T h e s e r ie s U n is s a id t o b e ( N , p n)-s u m m a b le t o s i f

n= о

1 "

1п = ~Б- Z Р п -к * к -+ s a s n - > o o ,

* n k = 0

w h e r e P n = p 0 + p t + . . . + p H, p 0 > 0 , p n ^ 0 , w e t h e n w r ite { N , p n) lim s„ = s o r { N , p n) Y j U n ^ s .

T h e s e q u e n c e {p „ } w ill b e s a id t o b e lo n g t o th e c la s s M “ fo r a c e r ta in r e a l a ^ 0 if

(i) 0 < p „ < p n + l fo r n = 0 , 1 , 2 , . . . o r 0 < p „ + l < p n fo r n = 0 , 1, 2 , . . . , (ii) P0 + P 1 + ••• + P n = Л ^ о о ,

(iii) lim ~ = a . n-»00 * n L et

Pn k = 0

Pk

k + l ‘

T h e s e q u e n c e { p n} w ill b e s a id t o b e lo n g t o th e c la s s B V M “ if { p n} e M a a n d if ( 5 „ | is a s e q u e n c e o f b o u n d e d v a r ia t io n , i.e.,

£ IS*- Sn-il < 00. n= 1

* Reader in Applied Mathematics, Dept, of Applied Mathematics, Faculty of Technology and Engineering, M. S. University of Baroda.

** U. G. C. Jr. Research Fellow, Dept, of Applied Mathematics, Faculty of Technology and Engineering, M. S. University of Baroda.

196 S. R. A graw al and P. S. K a n ta w a la

It is well known that the method (JV, Pn) is regular if and only if lim ^ = 0.

й-*00 P n

Obviously, if {pn} e M a, then the method (N, pn) is regular.

Let {Ф„(х)} (n = 0, 1, 2, ...) be an orthonormal system (ONS) of L2- integrable functions defined in the closed interval [a, b]. We consider the orthogonal series

GO

(1.1) X С„Ф„(Х)

n = 0

with real coefficients c„’s.

Let p(x) ^ x denote a positive function, concave from below, defined for x ^ 1 and increasing monotonically to infinity. We shall call the orthogonal series (1.1) p{n)-lacunary if the number of non-vanishing coefficients ck with n < к ^ 2n does not exceed p(n). Furthermore, we shall say that the coefficients have the positive number sequence {qn} as a majorant if the relation

(1.2) cn = 0(q„)

holds.

The sum

n b

K„(t, x) = z &k(t)&k(x) and Ln(x) = j | K n(t, x)\dt

k—0 a

are respectively called the n-th kernel and the n-th Lebesgue function of the ONS {Фп(х)}.

We denote as usual the n-th partial sums, (C, l)-means, (E , l)-means and (R , Xn, l)-means of the orthogonal series (1.1) by s„(x), <j„(x), Tn(x) and

<тп(Л, x), respectively.

Sunouchi [7] has discussed the convergence of

® |sn (x) (Tn (x)jfc

У --- , k > l ,

^ и

under the restriction of boundedness of the functions Фп{х).

Patel [5] has discussed the convergence of the series

(1.3) X

n — 1 and

(1.4)

OO I n= 1I 1'

\sn( x ) - T n(x)f

k W - M 'U ^)!

к ^ 2,

к > 2,

under the restriction of boundedness of the functions Фп(х).

The convergence of series (1.3) and (1.4) for к — 2 has been investigated by Meder [2] and Patel [4].

In this paper we prove analogous theorems for (N, p„)-summability where к = 2. We prove the following theorem:

Theorem 1. I f the coefficients o f the orthogonal series (1.1) satisfy the condition

(1.5) X c2n < °°

n= 0

and {pn} e M a, a ^ 0, then the series

»=i n

almost everywhere.

Further, we also discuss in this paper the (N, p„)-summability of the ,u(n)-lacunary orthogonal series (1.1).

Dealing with (C, a > 0)-summability of p{n)-lacunary orthogonal series (1.1), Alexits ([1], p. 130) has proved the following theorem:

Theorem A. I f the coefficients o f a p{n)-lacunary series (1.1) have as a majorant a positive monotone decreasing number sequence {q„} satisfying the condition

(1.6) I ^{s/ М Ч п}_n ^{< oo,}

then condition (1.5) implies the (C, a > 0)-summability almost everywhere of the orthogonal series (1.1).

The (E , g)-summability for q > 0 of the ^(ji)-lacunary orthogonal series (1.1) has been discussed by Sapre and Bhatnagar [6].

We extend in this paper the above results to (N , p„)-summability as follows:

Theorem 2. Let

(1.7) {p„}eBVAf\ a > i

and the coefficients of p(ri)-lacunary orthogonal series (1.1) have as a majorant a positive monotone decreasing number sequence {qn) satisfying conditions (1.5) and (1.6). Then series (1.1) is (N, pn)-summable almost everywhere.

In the above theorem we may exclude the condition of the lacunary property if we take into consideration that between the indices n and 2n there are exactly n free places; therefore, if we put p{x) — x, i.e., p(n) = n, then every row is p(n)-lacunary.

This remark enables us to point out a special case from Theorem 2 in which the condition of the lacunarity does not appear any more.

198 S. R. A graw al and P. S. K a n ta w a la

Theorem 3. I f the coefficients of the orthogonal series (1.1) have as a majorant a positive monotone decreasing sequence {q„} satisfying the condition

(1.8) ï 3 r < ° °

n= 1 Njn

and (1.7) holds, then the orthogonal series (1.1) is (N, p„)-summable almost everywhere.

Moreover, Alexits has discussed the (C, a > 0)-summability of the orthogonal series when the Lebesgue functions are uniformly bounded. More precisely, he has proved the following theorem:

Theorem B ([1], p. 185). I f the Lebesgue functions

(1.9) =

a k = 0

of an ONS {Ф„(х)} are uniformly bounded on the set E cz [a, b], then relation (1.5) implies the (C, a > 0)-summability o f the orthogonal series (1.1) almost everywhere on E.

We extend the Theorem В to Norlund summability as follows:

Theorem 4. I f the Lebesgue functions (1.8) of an ONS |Ф„(х)} are uniformly bounded on the set E c= [a, b], then the orthogonal series (1.1) is (N, p„)-summable almost everywhere on E under conditions (1.5) and (1.7).

In order to prove the above theorems we need the following lemmas:

Lemma 1 ([1], p. 118). Under condition (1.5) the relation sVn{ x )-o Vn{x) = ox{\)

is valid almost everywhere for every index sequence {rn| with vn+i/v„ > q > 1.

Lemma 2 ([3]). Let {nk} be an arbitrary increasing sequence of indices satisfying the condition

1 < q < nk+1/nk ^ r f o r k — 0 , 1 , 2 , . . . ,

where r and q are constants. I f conditions (1.5) and (1.7) hold, then the orthogonal series (1.1) is (N, pn)-summable almost everywhere if and only if the sequence (s„k(j>c)} is convergent almost everywhere.

P r o o f of T h e o re m 1. We have

n 1 ⁿ

s* (x )-t„(x )= X скФк( х ) - — £ pn. vsv

k = О * n t?= 0

1 ^{n n} 1 ^{n V}

= TT Z с*фк(*) X Pn-v— Б- X Pn-v X ck$k(x)

* n k = 0 t>=0 ” » | ! = 0 k= 0

J n n 1 n n

= X ^{Ск Ф к ( х)} £ P n - v — £ - Z ^{СкФЛ Х) ‘} Z ^{P n - v}

v = 0 ' лfc= 0 v = k

Л|к=0

к - 1 1 "

= 7rZ с*Ф_{Гяк= 0} *<х) Z_t>=0

Consequently,

(1.1°) £ - f(s„(* )-U * ))2^ = C* ( Z P*-*)2-

n= 1 n J п = 1 ПР п к = 0 0=o

If 0 < p „ f , then conditions (1.5) and (1.7) gives

b

Z “ f(s«(x)-Ux))2dx < Z ^~Б₂ Z

»=J«J » = l < k = 0

oo ao _ 2

- I *4 ’ I _{k = 1 n=k} ^{~р г}_{n r n}

■ = o ( D Z * 4 2 i i

k= 1 n = k ft

By B. Levy’s theorem

® (s„(x)-t„(*))2

= 0(1) Z c*< 00

k= 1

< oo almost everywhere in [а, Ь].

n=l «

If 0 < p„l, then (1.10) becomes

b

Z “ _Г [(sn{ x ) - t n{x))2^{d x ^} £ -L Z ^{cl k}2^Pl-k

1 n

w= 1 и =1nr nfc= 0

ao 00 _ 2

z *ч ² z

k = 1 и=к n r n

00 ao j

= 0 (1) £ к2сЦ X -3

k = 1 n = k n

= 0(1) X ck2 < 00.

k= 1

200 S. R. A graw al and P. S. K a n ta w a la

By B. Levy’s theorem

£ (5n(*)-fn(*))2 almost everywhere in [a, 6].

Thereby the theorem is completely proved.

P r o o f o f T h e o re m 2. Under conditions (1.5) and (1.6) Alexits has proved in Theorem A, the convergence almost everywhere of the sequence {^nW} of the partial sums of the orthogonal series (1.1).

Consequently, it follows by condition (1.7) and Lemma 2 that series (1.1) is (N, pj-summable almost everywhere.

P ro o f o f T h e o re m 3. Alexits ([1], p. 132) has proved that condition (1.8) implies condition (1.5).

Moreover, condition (1.8) is a special case of (1.6), corresponding to /л(п)

= n and this case is, as mentioned above, satisfied for every series. Hence our theorem follows from Theorem 2.

P ro o f of T h e o re m 4. From the given conditions we can conclude by Theorem В that the orthogonal series (1.1) is (C, a > 0)-summable almost everywhere on E.

Therefore, Lemma 1 implies the convergence of the sequence {s2„(x)} of partial sums of series (1.1).

Hence by Lemma 2 it follows that the orthogonal series (1.1) is (N, pn)~

summable almost everywhere on E.

We are greatly thankful to Dr. С. M. Patel, Reader in Applied Mathematics, Faculty of Technology and Engineering for his encouragement and valuable suggestion for the preparation of this paper.

References

[1] G. A le x it s , Convergence problems o f orthogonal series, Pergamon Press, 1961.

[2] J. M ed er, On the summability almost everywhere o f orthogonal series by the method of Euler-Knopp, Ann. Polon. Math. 5 (1958), 135-148.

[3] —, On the Norlund summability o f orthogonal series, ibidem 12 (1963), 231-256.

[4] С. M. P a te l, On a connection between Riesz and Euler means o f orthogonal series, Vikram Math. J. 1 (1966), 47-52.

[5] R. K. P a te l, Convergence and summability of orthogonal series, Ph. D. Thesis, M. S. U ni­

versity of Baroda, 1975.

[6] A. R. S a p re, and S. С. В h at n a gar, On the Euler summability o f lacunary orthogonal series, Vijnana Parishad Anusandhan Patrika 16 (1973), 203-206.

[7] G. S u n o u c h i, On the strong summability o f orthogonal series, Proc. Imp. Acad. Tokyo 20 (1944), 251-256.