IJMMS 2004:2, 55–64
PII. S0161171204301511
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
INCLUSION RESULTS FOR CONVOLUTION SUBMETHODS
JEFFREY A. OSIKIEWICZ and MOHAMMAD K. KHAN
Received 30 January 2003 and in revised form 9 June 2003
If B is a summability matrix, then the submethod Bλ is the matrix obtained by deleting a set
of rows from the matrix B. Comparisons between Euler-Knopp submethods and the Borel
summability method are made. Also, an equivalence result for convolution submethods is
established. This result will necessarily apply to the submethods of the Euler-Knopp, Taylor,
Meyer-König, and Borel matrix summability methods.
2000 Mathematics Subject Classification: 40C05, 40D25, 40G05, 40G10.
1. Introduction and notation. Let E be an infinite subset of N∪{0} and consider E as
the range of a strictly increasing sequence of nonnegative integers, say E := {λ(n)}∞
n=0 .
If B := (bn,k ) is a summability matrix, then the submethod Bλ is the matrix whose nkth
entry is Bλ [n, k] := bλ(n),k . Thus, for a given sequence x, the Bλ -transform of x is the
sequence Bλ x with
∞
bλ(n),k xk .
Bλ x n = (Bx)λ(n) :=
(1.1)
k=0
Since Bλ is a row submatrix of B, it is regular (i.e., limit preserving) whenever B is regular.
Row submatrices have appeared throughout the literature [5, 6, 8, 12], but they were
first studied as a class unto themselves by Goffman and Petersen [7], and later by Steele
[14]. The class of Cesàro submethods has been studied by Armitage and Maddox [1] and
Osikiewicz [11].
Let A and B be two summability matrices. If every sequence which is A-summable
is also B-summable to the same limit, then B includes A, denoted by A ⊆ B. Also, B is
called a triangle if bn,k = 0 for all k > n and bn,n ≠ 0 for all n. The following lemma
extends [1, Theorem 1].
Lemma 1.1. Let B be a summability matrix and let E := {λ(n)} and F := {ρ(n)} be
infinite subsets of N ∪ {0}.
(1) If F \ E is finite, then Bλ ⊆ Bρ .
(2) If B is a triangle and Bλ ⊆ Bρ , then F \ E is finite.
(3) If B is a triangle, then Bλ is equivalent to Bρ if and only if the symmetric difference
E F is finite.
In particular, B ⊆ Bλ for any λ.
Proof. Assume F \ E is finite and let x be a sequence that is Bλ -summable to L.
Then there exists an N such that {ρ(n) : n ≥ N} ⊆ E. That is, {ρ(n) : n ≥ N} is a
56
J. A. OSIKIEWICZ AND M. K. KHAN
subsequence of {λ(n)}. Since limn (Bλ x)n = limn (Bx)λ(n) = L, we have limn (Bρ x)n =
limn (Bx)ρ(n) = L.
Now assume B is a triangle, and hence invertible, and F \ E is infinite. Let F \ E :=
{ρ(n(j))}∞
j=0 with ρ(n(j)) < ρ(n(j + 1)). Consider the sequence y defined by
yk :=


(−1)j ,
if k = ρ n(j) for some j,

0,
otherwise,
(1.2)
and let x be the sequence B −1 y. Then, for every n,
Bλ x
n
= (Bx)λ(n) = B B −1 y λ(n) = yλ(n) = 0.
(1.3)
Hence, limn (Bλ x)n = 0. However, for every j,
Bρ x n(j) = (Bx)ρ(n(j)) = B B −1 y ρ(n(j)) = yρ(n(j)) = (−1)j .
(1.4)
Thus x is not Bρ -summable. Therefore Bρ does not include Bλ , which completes the
contrapositive of assertion (2). Lastly, assertion (3) follows from (1) and (2) since E F :=
(E \ F ) ∪ (F \ E).
To show the reason for the necessity of B being a triangle in assertion (2) of Lemma
1.1, consider the matrix B whose nkth entry is



0,






1,
B[n, k] :=



0,





1,
n
,
2
n
if n even and k = ,
2
if n odd and n ≠ k,
if n even and k ≠
(1.5)
if n odd and n = k.
Then if λ(n) := 2n and ρ(n) := 2n + 1, F \ E is infinite and Bλ ⊆ Bρ .
2. Inclusion results for Euler-Knopp submethods. For r ∈ C \ {0, 1}, the EulerKnopp method of order r is given by the matrix Er whose nkth entry is
 n k



r (1 − r )n−k ,
k
Er [n, k] :=


0,
if k ≤ n,
(2.1)
if k > n.
For the case r = 1, E1 is the identity matrix, and E0 is the matrix whose nkth entry is

1, if k = 0, n = 0, 1, 2, . . . ,
E0 [n, k] :=
0, otherwise.
It is well known that Er is regular if and only if 0 < r ≤ 1 (see [4]).
(2.2)
INCLUSION RESULTS FOR CONVOLUTION SUBMETHODS
57
Let E := {λ(n)} be an infinite subset of N∪{0} and r ∈ C\{0, 1}. The submethod Er ,λ
is the matrix whose nkth entry is


λ(n) k


r (1 − r )λ(n)−k , if k ≤ λ(n),

k
Er ,λ [n, k] :=



0,
if k > λ(n).
(2.3)
Then Er ,λ is regular if and only if Er is regular.
By a direct application of Lemma 1.1, we have the following inclusion result for the
Er ,λ methods.
Lemma 2.1. Let E := {λ(n)} and F := {ρ(n)} be infinite subsets of N ∪ {0} and r ≠ 0.
(1) The method Er ,λ ⊆ Er ,ρ if and only if F \ E is finite.
(2) The method Er ,λ is equivalent to Er ,ρ if and only if the symmetric difference E F
is finite.
We now examine the relationship between Er ,λ and the Borel summability method.
Recall that a sequence x is Borel summable to L if
lim e−t
t→∞
∞
k=0
xk
tk
= L.
k!
(2.4)
Theorem 2.2. Let E := {λ(n)} be an infinite subset of N ∪ {0} and r > 0. Then the
Borel summability method includes Er ,λ if and only if S := (N ∪ {0}) \ E is finite.
Proof. If S is finite, then by Lemma 2.1, Er and Er ,λ are equivalent. But the Borel
summability method includes Er for r > 0 (see [4]). Hence, it also includes Er ,λ . If S is
infinite, then it may be written as a strictly increasing sequence of nonnegative integers,
say S := {ρ(m)}∞
m=0 . If Mn := max 0≤k≤n |Er [n, k]|, consider the sequence y defined by

 ρ(m)! 2 ρ(m) + 1 Mρ(m) , if n = ρ(m),
yn :=
0,
otherwise,
(2.5)
and let x be the sequence Er−1 y; that is, y = Er x and
lim Er ,λ x n = lim Er x λ(n) = lim yλ(n) = 0.
n→∞
n→∞
n→∞
(2.6)
Hence, x is Er ,λ -summable to 0. Now observe that for a given n,
n
n
Er [n, k]
xk ≤ Mn
xk .
yn = Er x ≤
n
k=0
k=0
(2.7)
58
J. A. OSIKIEWICZ AND M. K. KHAN
Thus, for n = ρ(m), we have
1/ρ(m)
yρ(m) 1
1

·
·
ρ(m)! ρ(m) + 1 Mρ(m)

1/ρ(m)
ρ(m)
1
1
xk 
·
≤
.
ρ(m)! ρ(m) + 1 k=0

1/ρ(m)
=
ρ(m)!
(2.8)
Since lim supm (ρ(m)!)1/ρ(m) = ∞,

lim sup 
m→∞
1/ρ(m)
ρ(m)
1
1
·
= ∞,
xk 
ρ(m)! ρ(m) + 1 k=0
and it follows that lim supn (|xn |/n!)1/n = ∞. Thus,
nonzero t and hence x is not Borel summable.
∞
k=0 (xk /k!)t
(2.9)
k
diverges for all
Theorem 2.3. There exists a sequence which is Borel summable but not Er ,λ -summable for any λ and r > 0.
Proof. Let r > 0 and consider the sequence x defined by
2 n−1
1
1−
.
xn := n −
r
r
(2.10)
Then it can be shown that (Er ,λ x)n = (−1)λ(n) λ(n). Hence x is not Er ,λ -summable for
any λ. However,
e−t
∞
xk
k=0
∞
1
2 k−1 t k
tk
k −
= e−t
1−
k!
r
r
k!
k=1
∞
k−1
k
1 −t t
2
e
= −
1−
r
r
(k
−
1)!
k=1
∞
2 k tk
1
te−t
1−
= −
r
r
k!
k=0
1
= −
te−t e(1−2/r )t
r
1
te−(2/r )t .
= −
r
(2.11)
Since r > 0,
lim e−t
t→∞
∞
k=0
xk
tk
1
= lim −
te−(2/r )t = 0,
k! t→∞
r
and hence x is Borel summable to 0.
(2.12)
INCLUSION RESULTS FOR CONVOLUTION SUBMETHODS
59
3. Convolution methods. Let p and q be sequences of real numbers with pk ≥ 0,
∞
∞
qk ≥ 0, k=0 pk = 1, and k=0 qk = 1. The convolution summability method is given by
the matrix C ∗ := (cn,k ) whose nkth entry is



if n = 0,
qk ,



k
cn,k := 


 cn−1,j pk−j , if n ≥ 1.

(3.1)
j=0
∞
It is clear that C ∗ is a nonnegative matrix such that for every n, k=0 cn,k = 1. Some
classical summability matrices are examples of the matrix C ∗ . If 0 ≤ r ≤ 1, p := {1 −
r , r , 0, 0, . . .}, and q := {1, 0, 0, . . .}, then C ∗ is the Euler-Knopp method of order r . If
0 ≤ r < 1, p := {0, (1−r ), (1−r )r , (1−r )r 2 , . . .}, and q := {(1−r ), (1−r )r , (1−r )r 2 , . . .},
then C ∗ is the Taylor method of order r , denoted by Tr . If 0 < r < 1 and p := q :=
{(1−r ), (1−r )r , (1−r )r 2 , . . .}, then C ∗ is the Meyer-König method of order r , denoted
by Sr . If p := q := {1/k!e}, then C ∗ is the Borel matrix method B ∗ . Similar forms of the
convolution method are known by different names, such as the random-walk method
and Sonnenschein method. (Further information on all of these methods may be found
in [3, 4, 13].)
If C ∗ is the convolution method formed from the sequences p and q, then let
µ :=
∞
jpj ,
ν :=
j=0
∞
jqj .
(3.2)
j=0
We note here that for the remainder of this work, p and q are nonnegative sequences
whose sums are 1, and µ and ν represent the sums in (3.2). Also, cn,k := 0 whenever
k < 0.
We next present some preliminary results concerning the convolution method.
Lemma 3.1. The convolution method C ∗ is regular if and only if p0 < 1.
Proof. See [9].
Lemma 3.2. If µ < ∞ and ν < ∞, then for every n,
∞
kcn,k = nµ + ν.
(3.3)
k=0
Proof. Note that for n = 0, the result holds. So assume the result holds for some
integer n > 0. Then
∞
kcn+1,k =
k=0
∞
k=0
=
∞
j=0

k
k

cn,j pk−j  =
j=0

cn,j 
∞
∞
cn,j
j=0
ipi + j
i=0
By induction, the result follows.
∞
i=0

pi  =
∞
kpk−j
k=j
∞
j=0
µcn,j +
∞
j=0
(3.4)
jcn,j = (n + 1)µ + ν.
60
J. A. OSIKIEWICZ AND M. K. KHAN
Lemma 3.3. Let C ∗ be the convolution method formed from the sequences p and q and
:= {1, 0, 0, . . .}.
D := (dn,k ) the convolution method formed from the sequences p and q
Then for nonnegative integers n, k, and j,
∗
cn+j,k =
k
cn,k−i dj,i .
(3.5)
i=0
The proof of this lemma is a straightforward induction argument left to the reader.
Lemma 3.4. Let C ∗ be the convolution method formed from the sequences p and q.
∞
∞
If µ < ∞, ν < ∞, 0 < j=0 (j − µ)2 pj , and j=0 j 3 pj < ∞, then
∞
cn,k+1 − cn,k = O √1 .
n
k=0
(3.6)
Proof. Let D ∗ := (dn,k ) be the convolution method formed from the sequences p
:= {1, 0, 0, . . .}. We first prove that the result holds for D ∗ .
and q
√
∞
√
2
Let φ(t) := ( 2π et /2 )−1 and xn,k := (k − nµ)/σ n, where σ 2 := j=0 (j − µ)2 pj .
Then
√
n
∞
∞ 1
dn,k+1 − dn,k ≤ √n
dn,k+1 − √
φ
x
n,k+1
σ n
k=0
k=0
√
+ n
∞ 1
1
√ φ xn,k+1 − √
φ
x
n,k σ n
σ n
k=0
∞ √ + n
k=0
(3.7)
1
√ φ xn,k − dn,k .
σ n
The first and the third terms on the right-hand side of the inequality are bounded by a
result of Bikjalis and Jasjunas [2]. For the middle term, the mean value theorem yields
√
n
∞ ∞
1
1
1 φ ξn,k xn,k+1 − xn,k
√ φ xn,k+1 − √
φ xn,k =
σ n
σ
n
σ
k=0
k=0
K
φ (t)
dt < ∞,
<
σ R
(3.8)
where ξn,k ∈ (xn,k , xn,k+1 ) and K > 0 is some constant. Thus, the result holds for the
convolution method D ∗ . Then, by Lemma 3.3,
∞
∞ k+1
k
cn,k+1 − cn,k =
qk+1−i dn,i −
qk−i dn,i k=0
k=0 i=0
i=0
∞ k+1
k
qk+1−i dn,i −
qk−i dn,i =
qk+1 dn,0 +
k=0
i=1
i=0
INCLUSION RESULTS FOR CONVOLUTION SUBMETHODS
≤ p0n
∞
qk+1 +
k=0
k
∞ 61
qk−i dn,i+1 − dn,i k=0 i=0
∞
∞
dn,i+1 − dn,i qk−i
≤ pn +
0
i=0
k=i
∞
dn,i+1 − dn,i = O √1 .
= p0n +
n
i=0
(3.9)
4. Equivalence results for convolution submethods. Let E := {λ(n)} be an infinite
subset of N ∪ {0}. The convolution submethod Cλ∗ is the matrix whose nkth entry is
Cλ∗ [n, k] := C ∗ λ(n), k .
(4.1)
Lemma 4.1. The convolution submethod Cλ∗ is regular if and only if p0 < 1.
Proof. If p0 < 1, then C ∗ is regular and hence Cλ∗ is also regular. Conversely, if Cλ∗
∞
is regular and p0 = 1, then Cλ∗ [n, k] = qk for all n and k. Since k=0 qk = 1, there exists
such that q ≠ 0. Then limn C ∗ [n, k]
= q ≠ 0, which contradicts the regularity of C ∗ .
ak
k
λ
k
λ
The following theorem compares Cλ∗ with C ∗ for bounded sequences.
Theorem 4.2. Let C ∗ be the convolution method formed from the sequences p and q
∞
∞
with µ < ∞, ν < ∞, 0 < j=0 (j −µ)2 pj , and j=0 j 3 pj < ∞. Let E := {λ(n)} be an infinite
subset of N ∪ {0}. If
lim
n→∞
λ(n + 1) − λ(n)
= 0,
λ(n)
(4.2)
then C ∗ and Cλ∗ are equivalent for bounded sequences.
Proof. By Lemma 1.1, C ∗ ⊆ Cλ∗ for any λ. So assume limn (λ(n+1)−λ(n))/ λ(n) =
0 and let x be a bounded sequence that is Cλ∗ -summable to L. Consider the set S :=
{ρ(n)} := (N ∪ {0}) \ E. If S is finite, then Lemma 1.1 shows that Cλ∗ and C ∗ are equivalent for all sequences. So assume S is infinite. Then there exists an N such that for n ≥ N,
ρ(n) > λ(0). Since E and S are disjoint, for n ≥ N, there exists an integer m such that
λ(m) < ρ(n) < λ(m + 1). We write ρ(n) := λ(m) + j, where 0 < j < λ(m + 1) − λ(m).
Then, for n ≥ N,
∞
∞
∗ ∗ C x − C x =
cρ(n),k xk −
cλ(m),k xk ρ
n
m
λ
k=0
k=0
∞
∞
cλ(m)+j,k xk −
cλ(m),k xk =
.
k=0
k=0
(4.3)
62
J. A. OSIKIEWICZ AND M. K. KHAN
By Lemma 3.3, this becomes


∞ ∞
∞
∗ ∗ C x − C x =


c
d
−
c
x
x
λ(m),k−i j,i
k
λ(m),k k ρ
n
m
λ
k=0 i=0
k=0
 


∞
∞
∞
xk 
cλ(m),k−i dj,i  − 
cλ(m),k dj,i 
=
k=0
i=0
i=0
≤ x∞
∞
∞ dj,i cλ(m),k−i − cλ(m),k k=0 i=0
∞ i−1
dj,i
cλ(m),k−l − cλ(m),k−l−1 = x∞
i=0
k=0 l=0
∞
≤ x∞
∞
dj,i
i=0
(4.4)
∞ i−1
cλ(m),k−l − cλ(m),k−l−1 k=0 l=0
∞
i−1
x∞
cλ(m),k−l − cλ(m),k−l−1 .
dj,i
λ(m)
=
λ(m) i=0
l=0
k=0
∞
By Lemma 3.4, there exists an M > 0 such that
∞
cλ(m),k−l − cλ(m),k−l−1 < M.
λ(m)
(4.5)
k=0
Then, by Lemma 3.2,
∞
∞
i−1
∗ x∞ M x∞ M
C x − C ∗ x ≤ x∞
dj,i
M= idj,i ≤ · jµ.
ρ
n
m
λ
λ(m) i=0
λ(m) i=0
λ(m)
l=0
(4.6)
Since 0 < j < λ(m + 1) − λ(m),
∗ + 1) − λ(m)
C x − C ∗ x < x∞ Mµ · λ(m = o(1).
ρ
n
m
λ
λ(m)
(4.7)
Thus,
0 ≤ Cρ∗ x n − L
≤ Cρ∗ x n − Cλ∗ x m + Cλ∗ x m − L
= o(1) + o(1) = o(1).
(4.8)
Therefore, the sequence C ∗ x may be partitioned into two disjoint subsequences,
namely (Cλ∗ x)n = (C ∗ x)λ(n) and (Cρ∗ x)n = (C ∗ x)ρ(n) , each having the common limit L.
Thus, x must be C ∗ -summable to L, and hence C ∗ and Cλ∗ are equivalent for bounded
sequences.
The following theorem is a well-known result due to Meyer-König (see [10, Theorem 25]).
Theorem 4.3. The methods Er (0 < r < 1), Sr (0 < r < 1), Tr (0 < r < 1), and the
Borel method are equivalent for bounded sequences.
INCLUSION RESULTS FOR CONVOLUTION SUBMETHODS
63
Since the Euler-Knopp methods of order 0 < r < 1, Taylor methods of order 0 <
r < 1, Meyer-König methods of order 0 < r < 1, and the Borel matrix method all have
generating sequences satisfying the conditions in Theorem 4.2, the following corollary
is immediate.
Corollary 4.4. Let E := {λ(n)} be an infinite subset of N ∪ {0} and 0 < r < 1. If λ
satisfies condition (4.6), then Er ,λ , Er , Tr ,λ , Tr , Sr ,λ , Sr , Bλ∗ , B ∗ , and the Borel method are
all equivalent for bounded sequences.
The next theorem presents an equivalence relationship between the Cλ∗ submethods.
Theorem 4.5. Let C ∗ be the convolution method formed from the sequences p and
∞
∞
q with µ < ∞, ν < ∞, 0 < j=0 (j − µ)2 pj , and j=0 j 3 pj < ∞. Let E := {λ(n)} and
F := {ρ(n)} be infinite subsets of N ∪ {0}. If
lim
n→∞
ρ(n) − λ(n)
= 0,
λ(n)
(4.9)
then Cλ∗ and Cρ∗ are equivalent for bounded sequences.
Proof. Let x be a bounded sequence and consider the sequences M(n) := max{λ(n),
ρ(n)} and m(n) := min{λ(n), ρ(n)}. We write M(n) := m(n) + j, where j := M(n) −
m(n). For n ≥ 1, we have
∞
∞
∗ C x − C ∗x = cρ(n),k xk −
cλ(n),k xk ρ
n
n
λ
k=0
k=0
∞
∞
cM(n),k xk −
cm(n),k xk =
k=0
k=0
∞
∞
c
x
−
c
x
=
m(n)+j,k k
m(n),k k .
k=0
k=0
(4.10)
Then, as in the proof of Theorem 4.2, we have
∗ M(n) − m(n)
C x − C ∗ x ≤ O(1) j
= O(1)
ρ
n
n
λ
m(n)
m(n)
λ(n) ρ(n) − λ(n)
= O(1)
m(n)
λ(n)
(4.11)
= O(1) · O(1) · o(1) = o(1).
Then if x is Cλ∗ -summable to L,
0 ≤ Cρ∗ x n − L
≤ Cρ∗ x n − Cλ∗ x n + Cλ∗ x n − L
= o(1) + o(1) = o(1).
(4.12)
64
J. A. OSIKIEWICZ AND M. K. KHAN
Similarly, if x is Cρ∗ -summable to L, then
0 ≤ Cλ∗ x n − L
≤ Cρ∗ x n − Cλ∗ x n + Cρ∗ x n − L
= o(1) + o(1) = o(1).
(4.13)
Thus, Cλ∗ and Cρ∗ are equivalent for bounded sequences.
Acknowledgment. The authors would like to thank the referee for several helpful
suggestions that have improved the exposition of this work.
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Jeffrey A. Osikiewicz: Department of Mathematical Sciences, Kent State University, Tuscarawas
Campus, 330 University Dr. NE, New Philadelphia, OH 44663-9403, USA
E-mail address: josikiewicz@tusc.kent.edu
Mohammad K. Khan: Department of Mathematical Sciences, Kent State University, Kent, OH
44242-0001, USA
E-mail address: kazim@math.kent.edu