Internat. J. Math. & Math. Sci.
VOL. 12 N0. 2 (1989) 391-396
391
REGULARITY OF "F" METHOD OF SUMMABILITY OF SEQUENCES
YU CHUEN WEI
Department of Mathematics
Castleton State College
Castleton, VT
05735
(Received November Ii, 1987 and in revised form January 12, 1988)
ABSTRACT.
F
This paper is to develop theorems concerning the REGULARITY of the method
which is more general than
Cesaro’s, Able’s and Riemann’s methods In the
theory of summabillty.
KEY WORDS AND PHRASES.
Regularity of a Method of Summabillty of Sequences.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
40C15.
INTRODUCTION.
There are many well-knownmethods in the theory of summability which has many
uses throughout analysis and applied mathematics, for example,
Riemann’s, etc..
Cesaro’s, Able’s,
Mathematicians have contributed much to the study of these
methods which all can be found in books that provide an introduction to Summabillty
Theory.
The
"F"
method is one of methods of Summability, and more general than
But there is less information about Regularity available
those mentioned above.
covering the research in this method.
method.
2.
This note concerns the regularity of the
"F"
Five theorems will be given.
MAIN RESULTS.
DEFINITION 2.1.
an interval 0 < x
lim
x/0
Suppose that
{Fn(X) }n=
is a sequence of functions defined in
b and that for each n
I,
F (x)
n
and suppose that
F(x)
[
a
n
Fn(x)
is convergent in some interval 0 < x
lim F(x)
x/O
S.
c < b and
Y.C. WEI
392
[ an
Then we say that
is summable (F) to S.
[ an
It is not hard to see that if
a
n
is Cesaro or Abel or Riemann summable, then
is summable (F) for suitable functions F (x) respectively.
n
There is a well-known theorem about the regularity of the "F" method. [1,2]:
.
THEOREM.
(REGULARITY)
In order that the
"F"
method should be regular, it is
necessary and sufficient that
IFn (x) Fn+l (x)
< H
(2.1)
where H is independent of x, in some interval 0 < x
"F"
It is clear that method
is regular if {F
bounded in some interval 0 < x
c < b.
n
c < b.
(x)} is monotone and uniformly
Next first theorem will prove that the
"F"
method should be regular for some sequence of functions {F (x)} without monotonlcity.
n
The condition (2.1) is satisfied if there are two positive
THEOREM 2.1.
sequences
{mn }=On
{Mn }=On
and
<
0 < m
n
such that
for all n
mn+l
and
[
0<M
n
M
<
n
n=O
and for each n
m
-Fn(X)
PROOF.
11
<
MnX
0 < x
=<
c < b
Since
IFn (x) Fn+l (x)
<-we have
.
MnX
m
n
m
+
nM+IX
n+l
IFnCx) Fn+
m
=<
II
IFn+l(x)
I[ +
IFn(X)-
--<
(MnX
n
+
mn+
Mn+IX
=or
mo
Mox + 2 )
n=l
If 0 < x < r < I, then x
m
n
MnX
mn => xmn+l
.>
> O, and for any N
N
Y.
n=O
Mnl
--< A
(A is a constant)
For each such x, by Abel’s inequality
N
m
Y. MnX n[
n=l
m
m
-<- Ax I -<- Ar
REGULARITY OF "F" METHOD OF SUMMABILITY OF SEQUENCES
,
Let N
for any N.
we have
m
m
Thus
m
Let H
M0r
0
I MnX nl
n=l
.
.
Ar
1
m
IFn(x) Fn+ l(x)
<-
Mor
0
+ 2 Ar
m1
m
+ 2Ar
+ I.
H is independent of x and
IFn(x) Fn+ l(x)
in some interval 0 < x
< H
c < b.
Suppose "F" is regular.
THEOREM 2.2.
anF(x)
393
anFn(X)
Then
convergent implies that
convergent.
PROOF.
It follows from the regularity of "F" and lim Fn (x)
x/0
IFn(x) Fn+ l(x)
for each n, that
< H
and
IF0(x)
g
+ H
< H
(2.2)
n
are independent of x, in some interval 0 < x <- c < b.
where H, H
for any
IFn(x)
< H
> 0 and each x, we can choose
N-I
[.
anFn(X)
n--0
N >
No(e,x)
Let
F(x)
<
No(,x),
anFn(X)
F(x),
such that
,
+
n
S
also
[ aiFi(x)
n
i=0
P
P
I anFnZ(X)
N
[.
N
anFn(X)Fn(X)l
P
l (s=(x)
N
Sn_
(x))Fn(x)
P
I
[(Sn(x)
N
F(x)) + (F(x)
.
Sn_ l(x))]Fn(x)
P-I
<
IF( x
SN_ l(x)
IFN(X)
+
N
+
For P > N >
N0(e,x)
+ I,
it follows
[I Sn(x)
l(Sp(X)
-F(x)
F(x)
Fn(x)
IFp(X)
-Fn+ l(x)
Y.C. WEI
394
P
+ 3H),
N
Therefore,
anF(x)
Suppose that "F" is regular, then
COROLLARY 2.1.
that
anFnm(x)
PROOF.
k
F
that
c < b.
is a convergent in the interval 0 < x
anFn(X)
convergent implies
convergent, where m is a positive integer.
It follows from Theorem 2.2 that m
an n(X)
2 the assertion is true.
Suppose
is convergent and
k
[.
a F (x)
n n
G(x)
and let
k
n
[ aiF i(x)
i=O
S (x)
n
then
P
P
k+l
I anFn
N
k
I anFn(x)
N
(x)
Fn(x)
P
[
(S n(x)
N
Sn_ l(x)
Fn(x)
P
G(x)) + (G(x)
[(Sn(X)
N
Sn_ l(x))] Fn(x)
k+l
anFn
(x)
By the Axiom of Mathematical Induction, for all positive integer m
the assertion of the corollary is true.
THEOREM 2.3. Suppose that "F" is regular, then "Fm’’ is regular, where m is a
Repeating the procedure of the proof of Theorem 2.2, the convergence of
can be proved.
positive integer.
PROOF.
Since
m
m
IFn (x) Fn+l (x)
F (x)
n
(x) + F
n
(x)
Fn+l (x)
+
+
Fn+l (x)
(2.2) that
it follows from
m
and
IFn
Fn+l (x)
.
Fn(x)
<
m (H
m
Fn(X)
m-I
m
Fn+ l(x)
m
Fn+
m-I
+ H)
< m(H
.
Fn(X)
+ H)
Fn(X)
Fn+ l(x)
Fn+ l(x)
Hence for any positive integer m, if "F" is regular, then "Fm’’ is regular also.
REGULARITY OF "F" METIIOD OF SUMMABILITY OF SEQUENCES
Suppose that methods "F" and "G" are regular, then
THEOREM 2.4.
I)
F _+ G
2)
FG
3)
F
are regular;
is regular;
-I
is regular, if
inf
0<x__<c
(Fn(X)) ->
# 0.
e
all n
PROOF.
It follows from (2.1) and (2.2) that
[(Fn -+ Gn)
I)
<-- [ [(Fn
< [I
<
2)
+- (Gn
Fn+ I)
F
n
-F
+
(Fn+l
Gn+l
Gn+ 1)
+ Gn
n+
Gn+
HF+HG
IFnGn Fn+ Gn+
IFnGn Fn+IGn + Fn+iGn Fn+IGn+ll
[FG
n n
{Fn
< H
and
G
Fn+ Gn{
+
Fn+l{ {Gn{
IFn Fn+
IFnGn Fn+IGn+l
3)
[Fnl Fn+
IF: -F-I
n+l
.
{Gn
+
+ HF
[
and
Fn+ Gn Fn+ Gn+
[Gn
Gn+l{
Gn+
HFHG + HFIHG
[Fn+ Fn[
<
IFn+l{ {Fn{
IFn+
F
e
e-
[
<
H
Fn[
[Fn+
e
2
n
2
Fn+
(x)
n(x){
F
F
where H F, H G, H F, and H are independent of x, in some interval 0 < x
c < b.
G
Therefore, the assertions I), 2) and 3) are true. The proof is completed.
395
396
Y.C. WEI
REFERENCES
I.
GOLDBERG, Ro
2.
HARDY, G.H.
3.
POWELL, R.E. and SHAH, S.M. Summabilit Teory and its Applications, Van
Nostrand Reinhold Co., London, 1972.
4.
KNOPP, K. Theory and Application of Infinite Series, English translation by
R.C. Young, Blackie, London, 1928.
Methods of Real AnalTsis, John Wiley and Sons, Inc., New York, 1976.
Divergent Series, Oxford University Press, 1949.