I ntnat. J. Math. Math. Sci.
Vol. 3 No. 4 (1980) 731-738
SOME TAUBERIAN THEORENS FOR EULER AND BOREL SUMMABILITY
J. A. FRIDY
Department of Mathematics
Kent State University
Kent, Ohio 44252" U.S.A.
K. L. ROBERTS
Department of Mathematics
The University of Western Ontario
London, Ontario
Canada
(Received August 6, 1979 and in revised form December 7, 1979)
ABSTRACT.
The well-known summability methods of Euler and Borel are studied as
mappings from
11
into
1
I.
In this l- setting, the following Tauberian results
are proved: if x is a sequence that is mapped into
with r > 0
i
by the Euler-Knopp method E
or the Borel matrix method) and x satisfies
then x itself is in
En=01xn Xn+lln
r
< (R),
i.
KEY WORDS AN PHRASES. Taube condon, 1-1 method, Elr-Knopp mean, Borel
exponential mthod.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES:
Secondary
40D25.
Primary
40E05, 40G05, 40GI05
J.A. FRIDY AND K.L. ROBERTS
732
I.
INTRODUCTION.
In
[2,
p.
121], G. H. Hardy
described a Tauberian theorem as one which
asserts that a particular summability method cannot sum a divergent series that
oscillates too slowly.
In this paper we shall state the results in sequence-to-
sequence form, so a typical order-type Tauberian theorem for a method A would have
the form, "if x is a sequence such that Ax is convergent and
then x itself is
convergent." Our
AXk=Xk -Xk+ I o(),
the setting of ordinary convergence, but rather, we shall develop analogous
I
I.
- -
results for methods that map
into
method, and we shall henceforth write
that the matrix A determines an
Such a transformation is called an
I.
for
Ax/d
is in
c0,
JLER-KNOPP AND BOREL
The Euler-Knopp means
[6,
an
56-60]
pp.
n
0 < r _< I.
suPk
n=01ankl
4]
<-.
setting, it is necessary to
O(dk).
Since
analogue would be "Ax/d is in
.
k
are given by the matrix
(I- r)
n-k
0,
Theorem
Knopp and Lorent- proved
METHODS.
(k)r
[I,
,-,
=01AXkl/dk <
which we shall write in series form as
In
[5]
analogue of the above Tauberian condition Axk
this condition means that
2.
In
method if and only if
In order to prove Tauberian theorems in an
formulate an
r
Moreover, for such r,
if k--< n,
-
if k > n.
it is shown that E
.
determines an
E$1[] #
method if and only if
The customary form of Borel exponential summability is the sequence-tofunction transformation
if lim
t-m
-
present task is not to give more theorems in
([2,
p.
182], [6,
[e-=oXktk/k!]
-
p.
54])
given by
L, then x is Borel summable to L.
In order to consider this method in an
setting, we must modify it into a
,"
TAUBERIAN THEOREMS FOR EULER AND BOREL SUMMABILITY
733
This can be achieved by letting t tend to
sequence-to-sequence transformation.
Then B is the
through integer values and considering the resulting sequence Bx.
Borel matrix method
-
[6,
p.
56],
which is given by the matrix
b
e
-n k
n /k!.
By a direct application of the Knopp-Lorentz Theorem
an
matrix.
[5],
one can show that B is
We shall not use this direct approach, however, because the
assertion will follow from our first theorem, which is an inclusion theorem
between B and E
r
THEOREM i.
PROOF.
Since Bx
Since
EI
positive s.
If r > 0 and x is a sequence such that
ErX is
We use the familiar technique of showing that
(BEI)Erx,
El/r,
this will ensure that Bx is in
BEt
The n,k-th term of BE is given by
s
j-k k
s
s
( )(I
BEs[n’k] J=k e-nnJ
j! Jk
-n k k
e n s
n-k
k!
Y:’J=k (J-k)! (I s)j-k
(ns)ke -ns
k!
BEs,
we get
n=0 IBEs[n,k]l
1
n=0(ms)
o(Is).
k -ns
e
,
then Bx is
-- .
I
whenever
we replace I/r by s and show that BEs is an
Summing the k-th column of
in
is an
ErX
matrix.
is in
matrlx for all
J.A. FRIDY AND K.L. ROBERTS
734
Hence,
IBEs[n,k]l
SUPk n=0
< oo, so BEs
is an
-matrix.
Combining Theorem 1 with the knowledge that Er is an
following result as an immediate corollary.
THDREM 2.
-
The Borel matrix B determines an
matrix, we get the
method.
In addition to the inclusion relation given in Theorem I, we can show that
the
method B is strictly stronger than all E methods by the following
r
examp I e.
EXAMPLE.
.
(-s)
Suppose r > 0 and xk
but E x is not in
r
k,
where s k -I
+ 2/r;
then Bx is in
For,
(Bx) n
’k=0
k
-n
.( -s) k
(k 1(I
r
e
e
-n -sn
e
e
-n(s+l)
and
(ErX)n
n
=0
r’- rs <
By solving -I < 1
)n-k (-rs )k
I, we see that Erx
(I
is in
rs)
r
n
if and only if
-I < s < -I + 2/r.
We are now ready to prove the principal results which show that B and E can
r
not map a sequence from
into
if the sequence oscillates too slowly.
THDREM 3.
then x is in
PROOF.
If x is a sequence such that Bx is in
and
.
It suffices to show that Bx
Zn__01r__ 0 nkX- Xnl < ".
as
Znf01k=0 bnk(Xk Xn) l,
Since
k=0 bnk
x is in
;
that is,
I for each n, this sum can be witten
and so it suffices to show that
A=.nffi0k=0bnklXk-Xn ".
TAUBERIAN THEOREMS FOR EULER AND BOREL SUMMABILITY
We can write A
C
+ D, where
nl
n=Ok=0 bnklXk
C
x
and
.n=0k=n+l bnklXk Xn l"
D
Then
c
n-I
_<
==0
b
.n-I lrl
k =k
r
r=0 lxrln=r+l=0 bnk
.r=
IAXrlCr,
0
say.
Also,
D _<
k-i
T.n=0k=n+I bnk=n IAXrl
=0 AXrln=0<=r+l bnk
:o II,
By
me
L following, C
ich proves
L.
e
If
r
0()
and
say.
0),
so
theory.
bnk e-nnk/k
(1)
and r is a positive integer, then
Zn=r+l=O bnk
r
0 (),
n=0k=r+l bnk
0 ).
and
(il)
PROOF.
Let p
[],
r-p
r
nfr+l"k=0
If s < n, then (cf.
[2,
and write the sum in (i) as
bnk + nfr+l=r-p+l bnk Fr
p.
202])
+ or, say.
735
J.A. FRIDY AND K.L. ROBERTS
736
s
k
o
sss
+-+----+
n n
n
1
=0 k!
"-’)
s
(nS_)n 2 +
_<[n (i + s__+
"’’)
n
n
n
I
=r+l<+
p + I-
(n-rs>
In F r we have s
r
p and
max
n-r+p
rr+l
so
Fr <
In Gr we have
so
_
J + I)
=r
I
(r-p)’.
n=r+I
e
-n r-p _<
n
max
bnk =4 e
bnk <r lm
-p+l
’n=r+l
r
-n n
r
<--4.
n
e
+ I.
Hence, (i) is proved.
Next write the sum in (ii) as
_r+p-I
n=0k=r+l
(Assume that Hr
0 if p
n
I.)
Hr
-
< (p
n, then
k
n
+ Ir’ say.
Hr
Then
_<
If s
r
bnk + =0=r+p bnk
n
s
k
I)
e
=0
k>r
1
i)
n
k=s W. ( + s- +
n
s
<(
maX
n
=0
---[
e
-n r+l
n
n
n
s+t s+2 +
++ (s-y) 2 +’")
n
n
)
-
737
TAUBERIAN THEOREMS FOR EULER AND BOREL SUMMABILITY
n
s
7., (s +
Taking s
r
+ p,
s+l
we have
<
Ir<
r
n r+p
+p +
n=0 e- n (r .r+ p + I
I
(r+p)!
p.+
(r +p +’I
I)
I
(r+p).’
n
r
n r+p
n=0 e- n
_<+I.
This completes the proof of the L.
By combining Theorem 3 with Theorem I, we get an
the Euler-Knopp means.
.
THEOREM 4.
I
-
Tauberian theorem for
then x is in
If r > 0 and x is a sequence satisfying (*) such that E x is in
r
Next we give an application of these Tauberian theorems.
EXAMPLE.
by
Er,
The following sequence is not mapped into
with r > 0.
0
2
n /6
-I
k =x0-
--0
.
mk+l
Hence, by Theorem 3, Bx
.
is not in
Ax.
and
Then x satisfies (*), but x is not in
m
+(j+l)
-2
i/(j+l)2.
I,
because if k
Axj
-2
=--
=
m
I/k.
but much more tedious
sequence x such that Ax.
x is not in
or, afortiori,
Define x by
x
It is possible
by B
to construct a real number
.
and x changes sign infinitely many times, yet
For such an x, Theorem 3 implies that Bx cannot be in
J.A. FRIDY AND K.L. ROBERTS
738
REFERENCES
I.
Fridy, J. A., Absplute summability matrices that ar.e stronger than the
identity mapping, Proc. Amer. Math, Soc. 4__7(1975) 112-118.
2.
Hardy, G. H., Divergent Series, Clarendon Press, Oxford, 1949.
3.
Hardy, G. H. and J. E. Littlewood, Theorems concerning the summabilltT, of
series b7 Borel’s exponential method, Rendicontl Palermo, 4__1(1916) 36-53.
4.
Hardy, G. H. and J. E. Littlewood, On ..the Tauberlan theorem for
bility J. London Math. Soc. 1__8(1943) 194-200.
5.
Knopp, K. and G. G. Lorentz, Beitrag’4 zur
2_(1949) I0-16.
6.
absolute.n
.B.orel
Limltierung Arch. Math.
Powell, R. E. and S. M. Shah, Summability Theory and its Applications,
Van Nostrand Reinhold, London, 1972.