A N N A L E S S O C IE T A T IS M A T H E M A T IC A E PO LO N A E Series I : C O M M E N TA TIO N E S M A T H E M A T IC A E X X I I I (1983) R O C Z N IK I P O L S K IE G O T O W A R Z Y S T W A M A TEM A TY C ZN EG O
Sé ria I : P R A C E M A T EM A T Y C Z N E X X I I I (1983)
L u c y n a R e m p u l s k a (P o z n a n )
On the equivalence of logarithmic methods oï summability oî orthonormal series
1. Let
• oo
(1 ). ^ ^ ( a ? ) (а?е<0, 1»
* = 0
be a real orthonormal series with coefficients ck such that
O O
о
i.e. the orthonormal series of L 2. Denote by Sn(x) the wth partial sum of (1). Consider the following matrices X {<1) = ||X k (n, g)|| and the sequences
= {Y k (r, q)}tL0 (q = 1 ,2 ,3 ) : X 0( n , q ) = l ( g = 1 , 2 , 3 ) , '
■**(**, q) = 1
V - J -
U |- 1
p = 0
-1 к - 1 У — n + 1
p = 0
, n P -, v. k —1 P
H
p = 0 m —0
Jc n + 1
V у
X—i st—i m-\-1
p —0 m —0
if 3 = 1 ,
if q = 2 ,
if 3 = 3
(1 < к < n) n = 0 , 1, Х к{п, q) — 0 if к > n)j
r , ( r , î ) s 1 (2 = 1 , 2 , 3 ) ,
?k(r, 3) = 1 +
У rp+i Z l; «4-1 log (1 — T ) zZ p + 1
p = 0 k - 1
г ~ г Vr»+‘
log(l —r)'Z-J z Z p + 1 2
n = 0 JP = 0
, . k—l n p
t1 " 1-) y r»+i у у 1 log (1 — r ) z _ j Z_ j Z-i m + 1
71 = 0 J3 = 0 7 » = 0
if 3 = 1,
if g = 2 ,
if 3 = 3
(ft = 1 , 2 , . . . ; r e (0, 1 ));
We shall say that the series (1) is summable by the method X^q) (X{q)- summable) at xQ e <0, 1> to s if the sequence
П
(2 ) Z„(e, X<«>) ■**(», f l W ® )
A = 0
is convergent at x0 to s. We shall say that the series (1) is summable by the method Y{q) at x0 to s if
(3) P (x 0, r, Y (q))-^-s as r — >1 —, where
(4) P (x 0, r, Y(3)) = Yk (r,q)ek<pk (x0) for r e (0,1).
k = 0
The methods X (1) and Y(1) are called the logarithmic methods o f sum- mobility ([1 ], [3]); the method X (3) is the Cesàro method (G, 1).
Two methods of summability are called the equivalent methods in I?
if the summability of the series (1 ) in the set E of positive measure by one of these methods implies the summability of (1 ) to the same sum almost everywhere in P by the other method ([4]).
In this paper we shall prove that the methods X {q\ T (<z) are equivalent in L 2 for 3 = 1 and q = '2 .
2. First we shall give six lemmas on the summability of numerical series. Let Sn be the nth partial sum of the numerical series
00
' (5) 2 1"*
*=0
(uk are real numbers). The means L n(X^), P (r, Y(a)) and the summability we define as above. Clearly, the methods X {q) and Y {q) {q — 1 , 2 , 3) are the regular methods of summability of the series.
L emma 1. The series (5) is X^-summable to s i f and only i f it is X (3)-
summable to s.
Equivalence of logarithmic methods of summability 111
P roo f. Using the Abel transformation, we have
x/c=0 m=0 ' fc=о
* \-1 1 1
+<W+1)Z fcTTi,(Z<3,)l = A“+ 'B”’ *=o
n - 1 к .m к * + l . _ i
*•«">-v^SSS^ *+s|S^SZ *■*">+ fc=o m=o jj = o ' # = o p =o
» fc
+ГГ?тт( 7c=0 p = 0 ^ 7а_ П /r-i П * 7/. _ _ A /с — 0 2 тГ^|-<’-«-
' '
(n = 1, 2, ...). If-the sequence UW(X (3)) is convergent to s, then B n->s as n->oo,
and, by the Toeplitz theorem ([2], p. 427), An-> 0 as n-> oo.
If the sequence {Ln(X(2))} 'is convergent to s, then <7n->0 and B n->s as n->oo. Hence the lemma is proved.
L emma 2. Suppose that the series (5) is X^-summable to s. Then o{nlogn) i f q = 1,
o(n) i f q = 2 , 3 ; n - * o o . P roo f. We prove (6) for q = 1. We have
£ „ (V » )
( S Vc=0 m ) " S ' k —0 fc + 1 ■8„
Write
Clearly,
/ n+l-
in+ l(v > ) - £ „ = 2 ; — — 8n+1 'A = 0
and the sequence {Zn} is convergent to s if {Ln(X{1))} is convergent to s.
Hence (wlogw)"-1 Sn = o(l) and we have (6). Lemma 2 in the cases q — 2, 3 follows by Theorem 3 of [2], p. 522, and Lemma 1.
Z.
. n +1
- I S & + 1 V J _
Jfc + l
* = 0 s*-
L emma 3. I f the series (5) is summable to s by the method X (1) (X (2)), then it is summable to s by the method Y(1) (У(2)).
P ro o f. We shall give the proof for q = 2. From the definition of Yk (r, 2) we have
for any fixed r e (0,1). Hence the function P(r, Y(2)) is defined in (0, 1) and
00
F ir , Y « ) = j T Yt {r, 2 )uk
k ~ 0
.0 0
n
- t 1 - ^ y r»+i у 1 log(l — r) XLi Х- j 7г + 1
n = 0 fc=0
oo n к
_ -(Д--»-)2 y ^ + i у у
log(l — r) X- j Xu x L i
' n = 0 k —0 p = 0
1 P + 1
OO
s 3) - r „ +1(r, 3 ))i„ (V 2>).
n = 0
The convergence of sequence {Ln(X ^)} to s and the properties of Yk(r, 3) imply the existence of lim P (r, Y(2)) = s ([2], p. 513). The proof is com-
, r-> 1 —
pleted.
L emma 4. Suppose that q (q — 1, 2) is a fixed number. I f the series (5) is Y^-summable to s, then the series
(8) ] ? ( L k ( X ^ ) - L k_ ,(X ^ )) 0)
&-=0 is Y(s+1)-summable to s.
P ro o f. Let q — 1. Using the Abel transformation we have
(9) P (r, r “>) = j ^ ¥ t (r,l)<
* = 0
(1 - r ) oo к
У r*+i V —- — L k(X{1)) log(l — r) X— ■ 4 ' k=о j XLI p + 1 p=o r
OO
^ ( r t ( f ,2 ) - r w (r ,2 ) ) i l ( I l 1l) fc = 0
for r e (0,1). By (9) and (7),
Yn{r, 2 )L n(X {1))->0 as л-»оо, г e (0,1),
Equivalence of logarithmic methods of summability 113
and
oo
(10) P (r, Y<0) = 2 1 Xk (r, 2)(Lk( ^ ) - L t ^ (X ^ )]
k —o
(L ^ iX ^ ) = 0). The Y(1)-summability of (5) to s and (10) imply the Y(2)- summability of the series (8 ) to s. The proof for q = 2 is analogous.
L e m m a 5. I f the series (5) is Y^-summable to s and i f
oo
( 1 1 ) ^ ^ l o g ( f c + l ) < oo,
k = l
then (5) is convergent to s.
P roo f. Write
Y u lhlog(lc + l), rn — 1 --- - (n = 1, 2 , ...),
к=п n
If rn e (0, 1 ), then
n oo
Г <2,) - Я „ = Y ( r ‘ (*'».2 ) - l ) « t + У r t (r„,2)% s 7 „ + Z„.
By (11),
*=0 k = n + 1
|FJ<i s ^ j r ^ wt,oe(*+1) | lo g (l-rn)|
i - y »
| lo g (l-O I
( ™ \ 1/2 ^
| ^ ^ l o g ( ^ + l) | ^ lo g (fc + l ) | 1/2,
4=1 ' k=1
oo 1/2 00
^ r * ( r „ { 2 ^wogi^+i)}1'
к = И/ }С — 7Ъ
W n (± -rn) r i v v „ш+1 Z У — j j , + i î ' s |
|log(l-r„)|»log(» + l ) J \ £K£ i - P
w vW no OO 771 « 'no
2 >+и * 2 1 l d ~Iog(1~r)
p - t - 1 p _
/c= n m = A p = 0 m = n р = 0 Æ =w
"" fc
1 1
j?=o p + 1 \dr 1 — r 0 l | lo g (l-r„ )l\
l (1 - r nf
8 — R o c z n ik i P T M — P r a c e M a te m a ty c z n e t. X X I I I
Hence
\Vn\ = 0 { w n), \Zn\ = O(log~ll2n) and
( 1 2 ) IP {rn, T (2)) — $ J ->0 a s n-+co.
The Y(2h summability of the series (5) to s and (12) imply the conver
gence of the sequence {$n} to s. Thus the proof is completed.
Analogously we obtain
L e m m a 6 . I f the series (5) is Y^-summable to s and i f 00
Y.nu\ < oo, n = l
then (5) is convergent to s.
3. Now, we consider the summability of the series (1). We have
L e m m a 7. The series
CO
