Some criterions of Norlund summability

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I I I (1969)

J. Kopeć (Szczecin)

Some criterions of Norlund summability

Let 8 be the class of all real-valued functions bounded and inte- grable in each interval <0, a > (a > 0). We shall consider the linear method

oo

of summability Ta{ank} of the series (*) un whose coefficients are defined by the formula:

n = 0

(1) ank = exp j — j a{t)dĄ, n = 0 , 1 , & = ere 8 .

n—k

Write

П s

^ uk — the тг-th partial sum of series (*),

(2) k=0

tn = ankuk — the w-th Ta-mean of series (*).

k=0

Series (*) is said to be Ta-summable if the sequence {tn} is convergent.

It is clear, that for each positive sequence {Pn} there exists a function П

a (x )e⁸ such that Pn — P 0exp(J a{t)dt) (e.g. o{x) = L '(x), where у = L{x) о

is the equation of the polygonal line joining the points (n—1 , k i P ^ ) , П

(n, In P n), n = 1 , 2 , . . . ) . Hence if P n = p k > 0 {p0 = 1), then Pa-sum- k =0

inability and Norland (Ж, p n)-summability (where ank = Pn~klPn) are equivalent. In this note we shall show, that using form (1) of Norlund method it is possible to simplify several proofs and to get some generali­

zations of known theorems.

1. Regularity. It is well known (see e.g. [7], p. 127) that the (W, pn)~

method is regular if and only if the following conditions are satisfied:

lim Pn/Pn+i = 1, Ш = 0 {P n).

7c=0

(3)

I t follows from (3) that for TCT-method the following conditions of regularity hold.

1.1. Suppose that a{x)€S and write a_ (x) ⁼ min [(/(a?), 0]. I f

x-\-l со

lim J o(x)dx = 0 and f \a_{x)\dx < oo, then T„-method is regular.

x—нэо x 0

n +1

P ro o f. Since P n/Pm+1 = exp(— f cr(t)dt), the first condition (3) is П

satisfied. Writing a+{x) — тах[сг(ж), 0], we have for Tc = 1, 2 , . . .

к x

\Pk\ = \Pk~ Pft_il = I J o'(a?) exp (J a(t)dtj dx |

k-l о

к x

< J [<r+(a?)+|<r_(a?)|]exp(J a+(t)dt\ dx,

k-l 0

n n . Ж o o »

J" cr+ (a?)exp a+(t)dt)j dx-\- J |o"_ (a?)|da?exp|J* a+ (t)dtj

fc=l о ‘ о о о

n w.

= o j e x p j j a +(t)dt^ = 0 {e x p (J ⁰ (Pn).

о 0

П

1.2. I f Ta -method is regular, Йетг liminf J a{t)dt > — oo.

те—>oo 0

vn vn

Indeed, if lim J a{t)dt — — oo, then lim P Vji = 0 and £ \Pk\ > 1

и—>oо 0 n—>00 k —0

(p0 = 1). Therefore conditions (3) cannot be satisfied.

00

In particular if Pa-method is regular and J a+{t)d t< oo, then

oo 0

J |<7_ (£)|d< < oo.

0

2. Auxilliary lemmas. Suppose that ^{a e S} and that ^q is an arbitrary real number.

2.1. I f g ^ O and limsup(a?+l)1+ecr(a?) < oo, then there exists a con-

x—>oo

stant C > 0 {depending on q) such that

1—e x p ( — f a(t)dt)^ G '--- (0 1 J k 1 ( n + l ) ( * + l - f c ) e ^ Proof. I t follows from Bernoulli inequality that

1 - [ 1 - fc/(w+l)]g < 1 - [ 1 - Щ п + l)][g]+1 < — - - . n -\-1 Consequently, we have for q > 0

1Ш +.Р * -

«_* £ (w— J c+ lf q (w+l)(w—& + l)e

Therefore applying the mean-value theorem we obtain

ak \ к

1 —exp ( — J cr(/)dfój< 1 — expl ■I

n—k ' (n-\-l)(n—k-Ą-l)e < 0

( n + l) ( n — & + l)e and since a > 0, C is also a positive constant (depending on q). If q = 0, then a{x) < 5 ( # - j- l )-1 (В > 0) and

* n i -j } В

exp|— j a{t)dt^^\l- к

n - k ' '

n

Hence 1 —exp(— J a(t)dt) < (/^/(w+l).

n —k

2.2. I f q > 0 and liminf (x-\-l)l+e o(x) > — oo,

— >oo

&

1 —exp( — J a(t)dt^ > — Ox-

■ (n + l){n -\ -l-T c)e The proof is similar to that of 2.1.

2.3. I f limiiif (x-\-l)a(x) > then

(С г> 0).

n r

exp|— J o(t)dĄ = 0 \

n—k L

n —k-\- I T

---- — — ( k ^ n ) . n + 1 J

Proof. It follows from the assumption that there exists a constant X = X{fi) such that a(x) > jaf(x-j-l) for x > X. Let be n > X and 0 < к < n —X , thus

exp|— J a(t)dłj

n—k

n —k + i v n + 1 / * If n - X < к ^ щ then

Tb Jl lb / y’ i -1 \t*

exp|— J сг(£)<й| < exp|J \a{t)\dłj exp( — J a(t)dtj <

(Cx = (7x(-3^ “hi)** for ^/л > 0, (7X — Ox for ^{[j, < i} 0).

3. Numerical series. In this section we shall prove some theorems of tauberian type.

3.1. Assume that the function o(x )eS satisfies the following conditions:

(4) limsup (x -f 1) a (x) < o o, liminf (ж-f- 1)1+^<т(а?) > — oo

SC—>00 —>OQ

for some real /3 > 0.

n oo

I f lim (w+1)-1 £h\uk\ — 0 and the series £ un is Ta-summable, then

n—yoo к=1 n=0

it is convergent.

Proof. From (2) and (1) we have:

n n n

\tn—*nl < \ank—1| \uk\ = ^ | l — e x p (— J

k= 0 k=0 n—k (

Hence from 2.1 for q — 0 and from 2.2 for 0 < q < /3 we obtain П

\tn—sn\ = ⁰ { { n + l^{) - 1} ^ h \ u k\).

k—0

n

E e m a r k . The equality lim (w +l)-1 huk = 0 is the necessary and

łl—>-oo &=0 oo

sufficient condition of convergence of the (C, l)-summable series £ un

n = 0

(see [2], p. 524).

OO

3.2. I f <y(x) satisfies conditions (4), series (*) uk is Ta-summdble

oo к =0

and £ k u k < oo, then series (*) is convergent.

k = l

Proof. Cauchy inequality yields: t

n W 1 И

— Y'lc\uk\ = — — У &1 uk\-\--- — У Tc\uk\

4-1 jLj n -\-1 ^ n -\-1 .

n 4-1 _{k = l k = 1 k = N+ 1}

1/2

L 1 \fc=A7+ l ’ ^ -*=V+1wj-1 ’ J

Since

n

П

Ш

for N sufficiently large, the assumptions of 3.1 are satisfied and series (*) is convergent.

E e m a r k . Theorems 3.2 and 3.1 are generalizations of Lemma 5 of [4].

4. Summability of orthogonal series. Let the series

OO

( 5 ) ^ C n<pn(x)

n = 0

with coefficients {cn} e l2 be orthogonal in the interval < 0 , 1>. Moreover, let sn(x) be the n-th partial sum and let tn(x) be the w-th Ta-mean of series (5).

4.1. Suppose that the function a(x)ęS and that conditions (4) hold.

I f the sequence {nk} of indices satisfies the condition of lacunarity nk+1/nk

^ q > 1 (fc = 0 , 1 , ...), then the series OO

(A) — tnk(oc) ] 2

k=1 is convergent a.e. in <0 , 1> (x).

Proof. Prom (2) and (1) it follows

1 nk nk nk

j [8nk {so)- tnk(oc)]4x = £ [1 - ankijy c} = ^ [l — exp ( - j о {t)dtffc}.

0 j = 0 ? =0 nk- j

In the same way as in 2.1 we obtain

(A') J r * J [snfc {so) — tnk (ж)]2dx k=1 0

oo nk

'fc=1 7= l Since

2 PI(nk +\Y < у «*/(«*+*+ 1)* < У 3’ 2'

series (A') is convergent and according to Levi’s theorem series (A) is convergent a.e.

4.2. I f the function o(x )eS satisfies the conditions

(7) liminf{х + 1)в{х) > 1/2, limsup(a?+l)2|<T(a;)— o {x —1)| < o o,

Ж—^ОО «Г ) 00

then the series

OO

(B) ] ? n [ t n(x) — tn_ x{x)Y

n = 1

is convergent a.e.

Proof. From (2) we have

1 i n П— 1 4

J \tn{x) — tn_1{x)Ydx = § { ^ a nkCk<pk{x)— an-i,kCk<pk{x)f dx 0 * = 0

n—l

k=0

— ^ ‘ ®п_¹)к^ ск-\-anncn.

k=о

(x) It is a generalization of Theorem 3.1 of [3], where it was assumed that o{% )> 0 instead of the second condition (4).

Write в(х) = a {x —1) — a{x). It follows from (1) that

iV 71— 1 П

[ank — an_ 1>fc]2 = exp( —2 J cr(tf)dtf)|l —exp ( — J a(t)dt.+ f

n—k n—k—1 n—k

n n

= exp| —2 j a(t)dtj[l — e x p J 0(£)dż)J2.

n—k n—k

If [z< liminf(х-\-1)а(х) and limsnp(a?+l)1+e\в(х)\< oo > 0), then

X—K50 0С-УОО

from 2.3, 2.1 and 2.2 (for в instead of a) we obtain

\0/nk a n- i , k ] 2 —

Consequently

k 2(n — &+1)²/*—2q

(W + l)^{,2 ^ + 2} < n = 0 {(1 I+1)-2"}.

(8)

1 n

J lAO») — tn_x{x)Ydx = o j j T

о 'fc=l

k²( n - k + l f * ~ 2e (w+1)2^+2 4 l Putting q = 1 we have

(BO (•>]•«*» = o { £

4

n = l о v 4 = 1 k = l \ > / '

= О

oo 00

\ Z * * I

4 = 1 n = k+ 1

( n - T c + l f * - 2) ( n + l)2M+1 J For 1/2 < p < ± the function Фк{х) = х2/л 2/(a? + ft)2'“+1 is decreasing-

~ 1

for all к > 0 and x > 0 and J 0 k(x)dx = уу В(2/л—1, 2). Hence series

0 ™

(B') is convergent and series (B) is convergent a.e. in <0,1>.

B e m a r k . The assumptions of 4.2 are satisfied e.g. if e{x) = const

> 0, or if a(x) — C—lIx. In these cases the TCT-method is not regular..

X

If we suppose additionally that the condition of regularity lim f a{u)du

X-+OQ x —l

= 0 holds, then Theorems 4.2 and 3.2 of [3] are equivalent. Indeed, suppose П

that conditions (7) are satisfied, P n —exp(J a{t)dt), B n = pn/Pn = (Pn—

n ⁰

—P n-i)IP n = 1 — exp (—J a(t)dt). In [3] it was shown, that n(pn —

n—¹ n

—Vn-i)lVn = n 2(P n—B n ^ In R n + n R n ^ , B n—Bn_i = pn f d(u)du and

n n n— 1

fin = —exp(— f o(t)dt—ftn f 6(t)dt) (0 < # я < 1).

n -1 П—1

According to (7) and the additional assumption, we have 1 — exp (— J a(u)duj\\ = a?|J [o (t—1)—<r(<)]exp j a{u)du^dt

X — l X t— 1

oo

= 0|жJ |(т(<—1) — cr(i)|cfój = 0 ( 1 ) .

Therefore lim (3n = —1, w2 |i2n— j — 0(1), nBn = 0(1). More-

Ю—>-CO Ж

over, liminf > 1/2 since liminf# fa(u )d u > 1/2 . Hence {pn}e9JT and

П-УОО Х — УОО X — 1

a > 1/2 (see [4]). Conversely, if {рп}еЭДГ, у = L(x) is the equation of the polygonal line with tops in points (n, ln P n) (n = 0 , 1 ,. . .) , then putting a(x) — L' (x), n = [#] we have

( # + l)2\a{x—1) — a(x)\ < (тг + 2)2 In-Ł П+1Х 7)^{P P} P

-L m

{n-\- 2)2 In(-f*n “Ь^тг+ l ) (-^re P%

~ "p2-1 -и.

(n-f 2)2 In 1

0 (w-f 2)[j?n+1--;pJ (tt + 2)j»n+1

Pn+l Pn + (w + 2 )i?n+1(w +2 )j?,

P fi+l Pn PnPn + 1

Pr> P i

= 0(1).

I t is known ([3]) that in this case liminf (x-\-l)a(x) > a. Therefore

x->-oo

conditions (7) are satisfied.

4.3. Suppose that creS and liminf(#-f-l)cr(#) > 1/2, limsup(#-|-l)cr(#)

x—>oo X—>oo

< oo, limsup(# + l ) 2\a{x—1 ) — o(x)\< oo. Series (5) is Ta-summable a.e.

X —M X)

i f and only i f there exists an increasing sequence of indices {nk} such that 1 < q < nk+ilnk < r (q, r — constants) and the sequence {sn&(#)} is con­

vergent a.e. in <0 , 1>.

It follows from the assumptions that 4.1 and 4.2 hold. I t is well known ([4] and [3]) that 4.3 is a consequence of these theorems.

4.4. Suppose that a(x) satisfies the assumptions of 4.3 and that the increasing sequence o f indices {nk} satisfies the condition r > nk+ljnk > q > 1 (h = 0, 1 , ...). Moreover, let pn = max (it: nk < n] (k = 0 , 1 , . . . ; n0 = 1).

OO

I f the series JŚ/ c²nln²pn is convergent, Йе?г series (5) is Ta-summable in

71= 1

<0 , 1> a.e. (2)

Indeed, from the definition of pn it follows that pn+l > pn and

nk+1 °°

Pnk = k. Therefore, writing Ak = ( ^ d)lj2 we have £ Aklnzk =

v = nk +1 fe=2

(2) It is a generalization of Theorem 2 of [4].

oo nk + \ °o

— Cv ^ 2 f c2v\t\.2[i v < oo. Hence the sequence {snA.(a?)}

fc = 2 »»=n^. + l v= n 2+ l

is convergent a.e. in <0 , 1> (see [1 ], p. 79, Satz 2.3.4) and according to 4.3 series (5) is Ta-summable a.e.

4.5. Suppose that the function a{x )eS satisfies conditions (7) and let tfn{x) be the n-th Ta-mean of series (5). I f a1(x)eS satisfies conditions (4)

OO

and the series £ \fn{x) — C -i(#)] is Tai-summable a.e., then series (5) is

* П— 1

Ta-summable a.e.

Proof. Write rn(x) = tn{x) — tan_г{х), r⁰(x) = 0,й. = 1 , 2 , . . . It fol-

OO

lows from 4.2 that the series m2n(x) is convergent a.e. Moreover, the

oo n = 1

series 21 ^{r n{ % )} is PCTl-summable a.e. Theorefore, according to 3.2 it is

n =1

convergent a.e. Consequently series (5) is T0-summable a.e.

OO

5. Absolute Ty-summability. The series (*) 2! un is said to be

0 oo

absolutely Ta-summable (\Ta\-summable) if the series 2 \tn— tn_i\ is con-

n = l

vergent (tn is the w-th Тст-теап of series (*)). Suppose, that r < 1 is real and that the function ereS satisfies the following conditions:

lim inf(x-\-l)o(x) > 1/2 —r, Iimsup(a7 4- l ) 2_r|cr(a7—1) — a(x)\ < oo.

*C— ^oo 32——►OO

We shall consider the [TCT|-summability of the orthogonal series (5).

2W+1 oo

5.1. Write Am — ( 2 Tc2c l f 12. I f the series 2 2m(r l)Am is convergent, then series (5) is \Ta\-summable a.e.

Proof. It follows from (2 ), (1) and (8) for 1/2— r < liminf(a? + X-+CO -i-1) cr (at?) and q = 1 — r that

oo 2W+ 1 1 oo 2m + l 1

у 2 f l("(®)-*» -> (*) I < У {2” 2 ^11/2

m—0 n = 2m+ l 0 W =2m + 1

2 ^ + 1

= o { y [ 2” У у )ć2(«- ’>⁽ⁿ+ l) -2“- !d]1'2}

m=o n=2«t_|_i fc= 1

oo 2m+ * n

= 0 I ^ [ 2_<2P+1)“ 2

m—0 n=2m+l k = l

oo 2m+ 1 2m+ 1

= o j y [2- (2в+1>”* ( У й2с|) ( У

т=о fc = i n= 1

оо 2Ш-)-1 00

= о { у 2"<'-1) ( y k2cl)v) = 0 { у 2га<г-1Ыт} •

ft=l

and

o o 1 o o

(9) 8 = M®)-«^,(®)|d® = o j ^ 2m(r- 1)A m).

n = 2 0 m=Q

oo

Hence the series £ \tn(®) — tn-i(%)\ is convergent a.e.

n⁼²

2 ^ 4 ’ 1 o o

5.2. Write A'k = ( £ ^c2)1/2. I f the series £ is convergent,

7 = 2 fc+ l n = 1

tfrsw senes (5) is |Ta|-s^mma&Ze «.e.

Proof. Since

m 27 ' + l m m

^ = {o?+ % у v Ą yz« |Cli+ ( у 2и+! у « * Г < м + % 2f+lA't>

f—° * = rf+l 7 = ° * = 2? + l t=0

it follows from (9) that the following relation holds

o o oo m

8 = 0 { £ M + £ £ 2 f+1A'i\.

m = о m = o / = 0

Since r < 1, the first series is convergent. Changing the order of summation in t h e ' second sum of 8 we obtain

OO o o

8 = 0 { E 2>A'< = 0 { ^ 2 ,rA',}.

7 = 0 m > ? 7 = 0

E e m a r k . If {pn}eM a, a > 1/2, then the convergence of the series

OO

£ A'n is the necessary and sufficient condition of |JV, p n | - summability

n = l

proved in [5] (3).

References

[1] G-. A le x its , Eonvergenzprobleme der Orthogonalreihen, Budapest 1960.

[2] K. K n o p p , Szeregi nieskończone, Warszawa 1956.

[3] J . K o p e c, On some classes o f Nórlund means, Bui. Acad. Polon. Sci. 16 (2) (1968), pp. 93-98.

[4] J . Me der, On the N órlund summability o f orthogonal series, Ann. Polon. Matli.

12 (1963), pp. 231-256.

[5] — Absolute N órlund summability and orthogonal series, ibidem 18 (1966), pp. 1-13.

[6] — Some theorems on the N órlund summability o f orthogonal series, Bull. Acad.

Polon. Sci. 16 (3) (1968), pp. 201-207.

[7] K. Z e lle r, Theorie der Lim itierungsverfahren, Berlin 1958.

(3) In the note [6] Prof. Meder proved later that this theorem is valid also if { Pn} е$Ша and 0 < lim sup

n —> oo

n \p n — p n -l\

Pn — в < a — liminfnpn