Inclusion results for convolution submethods

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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

IfBis a summability matrix, then the submethodBλis the matrix obtained by deleting a set of rows from the matrixB. 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. LetEbe an inﬁnite subset ofN∪{0}and considerEas the range of a strictly increasing sequence of nonnegative integers, sayE:= {λ(n)}∞

n=0.

IfB:=(bn,k)is a summability matrix, then the submethodBλis the matrix whosenkth entry isBλ[n,k]:=bλ(n),k. Thus, for a given sequencex, theBλ-transform ofxis the sequenceBλxwith

Bλx

n=(Bx)λ(n):= ∞

k=0

bλ(n),kxk. (1.1)

SinceBλis a row submatrix ofB, it is regular (i.e., limit preserving) wheneverBis 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].

LetAandB be two summability matrices. If every sequence which isA-summable is alsoB-summable to the same limit, thenB includesA, denoted byA⊆B. Also,B is called a triangle ifbn,k=0 for allk > nandbn,n≠0 for alln. The following lemma extends [1, Theorem 1].

Lemma1.1. LetB be a summability matrix and letE:= {λ(n)}andF:= {ρ(n)}be inﬁnite subsets ofN∪{0}.

(1)IfF\Eis ﬁnite, thenBλ⊆Bρ.

(2)IfBis a triangle andBλ⊆Bρ, thenF\Eis ﬁnite.

(3)IfBis a triangle, thenBλis equivalent toBρif and only if the symmetric diﬀerence EFis ﬁnite.

In particular,B⊆Bλfor anyλ.

(2)

subsequence of{λ(n)}. Since limn(Bλx)n=limn(Bx)λ(n)=L, we have limn(Bρx)n= limn(Bx)ρ(n)= L.

Now assumeB is a triangle, and hence invertible, and F\E is inﬁnite. LetF\E:= {ρ(n(j))}∞

j=0withρ(n(j)) < ρ(n(j+1)). Consider the sequenceydeﬁned by

yk:=     

(−1)j_, _if_k₌_ρ_n(j)_{for some}_j_,

0, otherwise,

(1.2)

and letxbe the sequenceB−1y. Then, for everyn,

Bλxn=(Bx)λ(n)=BB−1yλ(n)=yλ(n)=0. (1.3)

Hence, limn(Bλx)n=0. However, for everyj,

Bρxn(j)=(Bx)ρ(n(j))=BB−1yρ(n(j))=yρ(n(j))=(−1)j. (1.4)

Thus xis notBρ-summable. Therefore Bρ does not includeBλ, which completes the contrapositive of assertion (2). Lastly, assertion (3) follows from (1) and (2) sinceEF:=

(E\F)∪(F\E).

To show the reason for the necessity ofBbeing a triangle in assertion (2) ofLemma 1.1, consider the matrixBwhosenkth entry is

B[n,k]:=                   

0, ifneven andk≠n

2, 1, ifneven andk=n₂,

0, ifnodd andn≠k,

1, ifnodd andn=k.

(1.5)

Then ifλ(n):=2nandρ(n):=2n+1,F\Eis inﬁnite andBλ⊆Bρ.

2. Inclusion results for Euler-Knopp submethods. For r ∈ C\ {0,1}, the Euler-Knopp method of orderris given by the matrixEr whosenkth entry is

Er[n,k]:=       

n k r

k₍₁₋_{r )}n−k_, _if_k_≤_n_,

0, ifk > n.

(2.1)

For the caser=1,E1is the identity matrix, andE0is the matrix whosenkth entry is

E0[n,k]:=   

1, ifk=0,n=0,1,2,...,

0, otherwise. (2.2)

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LetE:= {λ(n)}be an inﬁnite subset ofN∪{0}andr∈C\{0,1}. The submethodEr ,λ is the matrix whosenkth entry is

Er ,λ[n,k]:=         

λ(n)

k r

k₍₁₋_{r )}λ(n)−k_, _if_k_≤_λ(n)_,

0, ifk > λ(n).

(2.3)

ThenEr ,λis regular if and only ifEr is regular.

By a direct application ofLemma 1.1, we have the following inclusion result for the

Er ,λmethods.

Lemma2.1. LetE:= {λ(n)}andF:= {ρ(n)}be inﬁnite subsets ofN∪{0}andr≠0. (1)The methodEr ,λ⊆Er ,ρif and only ifF\Eis ﬁnite.

(2)The methodEr ,λis equivalent toEr ,ρif and only if the symmetric diﬀerenceEF is ﬁnite.

We now examine the relationship betweenEr ,λand the Borel summability method. Recall that a sequencexis Borel summable toLif

lim t→∞e

−t∞

k=0 xkt

k

k! =L. (2.4)

Theorem2.2. LetE:= {λ(n)}be an inﬁnite subset ofN∪ {0}andr >0. Then the Borel summability method includesEr ,λif and only ifS:=(N∪{0})\Eis ﬁnite.

Proof. IfS is ﬁnite, then byLemma 2.1,E_r andE_{r ,λ} are equivalent. But the Borel summability method includesEr forr >0 (see [4]). Hence, it also includesEr ,λ. IfS is inﬁnite, then it may be written as a strictly increasing sequence of nonnegative integers, sayS:= {ρ(m)}∞

m=0. IfMn:=max0≤k≤n|Er[n,k]|, consider the sequenceydeﬁned by

yn:=   

ρ(m)!2ρ(m)+1Mρ(m), ifn=ρ(m),

0, otherwise, (2.5)

and letxbe the sequenceE−1r y; that is,y=Erxand

lim n→∞

Er ,λxn=_nlim_→∞

Erxλ(n)=_nlim_→∞yλ(n)=0. (2.6)

Hence,xisEr ,λ-summable to 0. Now observe that for a givenn,

_y_n₌_E_r_x n≤

n

k=0

_E_r_[n,k]_x_k_≤_M_nn k=0

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Thus, forn=ρ(m), we have

ρ(m)!1/ρ(m)= 

 1

ρ(m)!· 1

ρ(m)+1·

_y_ρ(m)

Mρ(m)  

1/ρ(m)

≤ 

 1

ρ(m)!· 1

ρ(m)+1 ρ(m)

k=0

_x_k

1/ρ(m) .

(2.8)

Since lim sup_m(ρ(m)!)1/ρ(m)_{= ∞}_,

lim sup m→∞



 1

ρ(m)!· 1

ρ(m)+1 ρ(m)

k=0

_x_k

1/ρ(m)

= ∞, (2.9)

and it follows that lim sup_n(|xn|/n!)1/n = ∞. Thus, ∞k=0(xk/k!)tk diverges for all nonzerotand hencexis not Borel summable.

Theorem 2.3. There exists a sequence which is Borel summable but notE_{r ,λ} -sum-mable for anyλandr >0.

Proof. _Let_{r >}_{0 and consider the sequence}_x_{deﬁned by}

xn:=n

−1

r

1−2

r

n−1

. (2.10)

Then it can be shown that(Er ,λx)n=(−1)λ(n)λ(n). Hencexis notEr ,λ-summable for anyλ. However,

e−t ∞

k=0 xkt

k

k!=e

−t∞

k=1

k

−1

r

1−2

r

k−1_tk

k!

=

−1

r

e−t ∞

k=1

1−2

r

k−1 tk (k−1)!

=

−1

r

te−t ∞

k=0

1−2

r

k_tk

k!

=−1

r

te−t_e(1−2/r )t

=−1

r

te−(2/r )t.

(2.11)

Sincer >0,

lim t→∞e

−t∞

k=0 xkt

k

k!=tlim→∞

−1

r

te−(2/r )t=0, (2.12)

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3. Convolution methods. Letpand q be sequences of real numbers withpk≥0, qk≥0,∞k=0pk=1, and∞k=0qk=1. The convolution summability method is given by the matrixC∗:=(cn,k)whosenkth entry is

cn,k:=           

qk, ifn=0, k

j=0

cn−1,jpk−j, ifn≥1.

(3.1)

It is clear thatC∗is a nonnegative matrix such that for everyn,∞_k₌0cn,k=1. Some classical summability matrices are examples of the matrixC∗. If 0≤r ≤1,p:= {1−

r ,r ,0,0,...}, and q:= {1,0,0,...}, thenC∗ is the Euler-Knopp method of orderr. If 0≤r <1,p:= {0,(1−r ),(1−r )r ,(1−r )r2_,..._}_{, and}_q_:_{= {}₍₁₋_{r ),(}₁₋_{r )r ,(}₁₋_{r )r}2_,..._}_,

thenC∗ is the Taylor method of orderr, denoted byTr. If 0< r <1 andp:=q:= {(1−r ),(1−r )r ,(1−r )r2_,..._}_{, then}_C∗_{is the Meyer-König method of order}_r_{, denoted}

bySr. Ifp:=q:= {1/k!e}, thenC∗is the Borel matrix methodB∗. 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].)

IfC∗is the convolution method formed from the sequencespandq, then let

µ:= ∞

j=0

jpj, ν:= ∞

j=0

jqj. (3.2)

We note here that for the remainder of this work,pandqare nonnegative sequences whose sums are 1, andµ andν 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_,

∞

k=0

kcn,k=nµ+ν. (3.3)

Proof. _{Note that for}_n=_{0, the result holds. So assume the result holds for some}

integern >0. Then

∞

k=0

kcn+1,k= ∞

k=0 k

 k

j=0

cn,jpk−j  ₌∞

j=0 cn,j

∞

k=j kpk−j

=

∞

j=0 cn,j

 ∞

i=0 ipi+j

∞

i=0 pi

 ₌∞

j=0 µcn,j+

∞

j=0

jcn,j=(n+1)µ+ν.

(3.4)

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Lemma3.3. LetC∗be the convolution method formed from the sequencespandqand D∗:=(dn,k)the convolution method formed from the sequencespandq:= {1,0,0,...}. Then for nonnegative integersn,k, andj,

cn+j,k= k

i=0

cn,k−idj,i. (3.5)

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_. Ifµ <∞,ν <∞,0<∞j=0(j−µ)2pj, and∞j=0j3pj<∞, then

∞

k=0

_c_n,k₊₁₋_c_n,k₌_O 1 √

n

. (3.6)

Proof. _Let_D∗_:=_(d_n,k₎_{be the convolution method formed from the sequences}_p andq:= {1,0,0,...}. We ﬁrst prove that the result holds forD∗.

Letφ(t):=(√2πet2_/₂

)−1 _and _x

n,k:=(k−nµ)/σ√n, whereσ2:=∞j=0(j−µ)2pj. Then

√

n ∞

k=0

_d_n,k₊₁₋_d_n,k_≤√

n ∞

k=0

dn,k+1− 1 σ√nφ

xn,k+1

+√n ∞

k=0

_σ√1

nφ

xn,k+1

− 1

σ√nφ

xn,k

+√n ∞

k=0

_σ√1

nφ

xn,k

−dn,k .

(3.7)

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 ∞

k=0

_σ1√

nφ

xn,k+1−

1

σ√nφ

xn,k= 1

σ ∞

k=0

_φ_ξ

n,kxn,k+1−xn,k

<K σ

R

_φ_(t)_{dt <}_∞_,

(3.8)

whereξn,k∈(xn,k,xn,k+1)andK >0 is some constant. Thus, the result holds for the convolution methodD∗. Then, byLemma 3.3,

∞

k=0

_c_n,k₊₁₋_c_n,k₌∞ k=0

k+1

i=0

qk+1−idn,i− k

i=0 qk−idn,i

=

∞

k=0

qk+1dn,0+ k+1

i=1

qk+1−idn,i− k

i=0 qk−idn,i

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≤pn0 ∞

k=0 qk+1+

∞

k=0 k

i=0

qk−idn,i+1−dn,i

≤pn 0+

∞

i=0

_d_n,i₊₁₋_d_n,i∞ k=i

qk−i

=pn0+ ∞

i=0

_d_n,i₊₁₋_d_n,i₌_O 1 √

n

.

(3.9)

4. Equivalence results for convolution submethods. LetE:= {λ(n)}be an inﬁnite subset ofN∪{0}. The convolution submethodCλ∗is the matrix whosenkth entry is

Cλ∗[n,k]:=C∗

λ(n),k. (4.1)

Lemma_4.1. _{The convolution submethod}_C∗

λ is regular if and only ifp0<1.

Proof. _If_p0_<_{1, then}_C∗_{is regular and hence}_C∗

λ is also regular. Conversely, ifCλ∗ is regular andp0=1, thenC_λ∗[n,k]=qkfor allnandk. Since∞k=0qk=1, there exists aksuch thatqk≠0. Then limnCλ∗[n,k] =qk≠0, which contradicts the regularity ofCλ∗.

The following theorem comparesCλ∗withC∗for bounded sequences.

Theorem4.2. LetC∗be the convolution method formed from the sequencespandq withµ <∞,ν <∞,0<∞j=0(j−µ)2pj, and∞j=0j3pj<∞. LetE:= {λ(n)}be an inﬁnite subset ofN∪{0}. If

lim n→∞

λ(n+1 )−λ(n)

λ(n) =0, (4.2)

thenC∗andCλ∗are equivalent for bounded sequences.

Proof. _By_{Lemma 1.1}_,_C∗⊆_C∗

λ for anyλ. So assume limn(λ(n+1)−λ(n))/

λ(n)= 0 and letx be a bounded sequence that isCλ∗-summable toL. Consider the setS:= {ρ(n)}:=(N∪{0})\E. IfSis ﬁnite, thenLemma 1.1shows thatCλ∗andC∗are equiva-lent for all sequences. So assumeSis inﬁnite. Then there exists anNsuch that forn≥N,

ρ(n) > λ(0). SinceEandSare disjoint, forn≥N, there exists an integermsuch that

λ(m) < ρ(n) < λ(m+1). We writeρ(n):=λ(m)+j, where 0< j < λ(m+1)−λ(m). Then, forn≥N,

_C∗ ρx

n−

Cλ∗x

m=

∞

k=0

cρ(n),kxk− ∞

k=0

cλ(m),kxk

=

∞

k=0

cλ(m)+j,kxk− ∞

k=0

cλ(m),kxk .

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ByLemma 3.3, this becomes _C∗ ρx n−

Cλ∗x

m= ∞ k=0  ∞ i=0

cλ(m),k−idj,i  _x_k₋∞

k=0

cλ(m),kxk = ∞

k=0 xk

   ∞

i=0

cλ(m),k−idj,i  ₋

 ∞

i=0

cλ(m),kdj,i    

≤ x∞ ∞

k=0 ∞

i=0

dj,icλ(m),k−i−cλ(m),k

= x∞ ∞

i=0 dj,i

∞

k=0

i−1

l=0

cλ(m),k−l−cλ(m),k−l−1

≤ x∞ ∞

i=0 dj,i

∞

k=0 i−1

l=0

_c_λ(m),k₋_l₋_c_λ(m),k₋_l₋₁

=x∞

λ(m) ∞

i=0 dj,i

i−1

l=0

λ(m) ∞

k=0

_c_λ(m),k₋_l₋_c_λ(m),k₋_l₋₁_.

(4.4)

ByLemma 3.4, there exists anM >0 such that

λ(m) ∞

k=0

_c_λ(m),k₋_l₋_c_λ(m),k₋_l₋₁_{< M.} (4.5)

Then, byLemma 3.2,

_C∗ ρx

n−

Cλ∗x

m≤ x∞

λ(m) ∞

i=0 dj,i

i−1

l=0

M=x∞M

λ(m) ∞

i=0

idj,i≤x∞M

λ(m) ·jµ. (4.6)

Since 0< j < λ(m+1)−λ(m),

_C∗ ρx

n−

Cλ∗x

m<x∞Mµ·

λ(m+1)−λ(m)

λ(m) =o(1). (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 limitL. Thus,xmust beC∗-summable toL, and henceC∗andCλ∗are equivalent for bounded sequences.

The following theorem is a well-known result due to Meyer-König (see [10, Theo-rem 25]).

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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 inTheorem 4.2, the following corollary is immediate.

Corollary4.4. LetE:= {λ(n)}be an inﬁnite subset ofN∪ {0}and0< r <1. Ifλ satisﬁes condition (4.6), thenEr ,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 theCλ∗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)2pj, and ∞j=0j3pj<∞. Let E:= {λ(n)}and F:= {ρ(n)}be inﬁnite subsets ofN∪{0}. If

lim n→∞

ρ(n) −λ(n)

λ(n) =0, (4.9)

thenCλ∗andCρ∗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 writeM(n):=m(n)+j, wherej:=M(n)−

m(n). Forn≥1, we have

_C∗ ρx

n−

Cλ∗x

n=

∞

k=0

cρ(n),kxk− ∞

k=0

cλ(n),kxk

=

∞

k=0

cM(n),kxk− ∞

k=0

cm(n),kxk

=

∞

k=0

cm(n)+j,kxk− ∞

k=0

cm(n),kxk .

(4.10)

Then, as in the proof ofTheorem 4.2, we have

_C∗ ρx

n−

Cλ∗x

n≤O(1) j

m(n)=O(1)

M(n) −m(n) m(n)

=O(1)

λ(n) m(n)

_ρ(n)₋_λ(n)

λ(n)

=O(1)·O(1)·o(1)=o(1).

(4.11)

Then ifxisCλ∗-summable toL,

0≤Cρ∗x

n−L≤Cρ∗x

n−

Cλ∗x

n+Cλ∗x

n−L

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Similarly, ifxisCρ∗-summable toL, 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∗andCρ∗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:[email protected]

Mohammad K. Khan: Department of Mathematical Sciences, Kent State University, Kent, OH 44242-0001, USA