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On double Hausdorff summability method

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Abstract

Das (Proc. Camb. Philos. Soc. 67:321-326, 1970) proved that every conservative Hausdorff matrix is absolutely k th power conservative. Savaş and Rhoades (Anal. Math. 35:249-256, 2009) proved the result of Das for double Hausdorff summability. In this paper we will consider the double Endl-Jakimovski (E-J) generalization and we will prove the corresponding result of Savaş and Şevli (J. Comput. Anal. Appl. 11:702-710, 2009) for double E-J generalized Hausdorff matrices.

MSC:40F05, 40G05.

Introduction and background

The basic theory of Hausdorff transformations for double sequences was developed by Adams [1] in 1933. Later a few authors studied double Hausdorff matrices; see e.g. Ramanujan [2] and Ustina [3].

Several generalizations of Hausdorff matrices have been made. One of them is the Endl-Jakimovski, or E-J generalization defined independently by Endl [4] and Jakimovski [5] as follows.

Let β be a real number, let ( μ n ) be a real sequence, and let Δ be the forward difference operator defined by Δ μ k = μ k μ k + 1 , Δ n ( μ k )=Δ( Δ n 1 μ k ). Then the infinite matrix ( H ( β ) , μ n ( β ) )=( H β ,μ)=( h n k ( β ) ) is defined by

h n k ( β ) = { ( n + β n k ) Δ n k μ k ( β ) , 0 k n , 0 , k > n ,

and the associated matrix method is called a generalized Hausdorff matrix and generalized Hausdorff method, respectively. The moment sequence μ n ( β ) is given by

μ n ( β ) = 0 1 t n + β dχ(t),

where χ(t)BV[0,1]. The case β=0 corresponds to ordinary Hausdorff summability.

In a recent paper [6], the first author jointly with Savaş has extended the result of Das [7] to the E-J matrices; i.e., all conservative E-J matrices are absolutely k th power conservative for k1. Thereafter, Savaş and Rhoades [8] proved the result of Das [7] for double Hausdorff summability. In this paper we will consider double E-J generalization and we will prove the corresponding result of [6] for double E-J generalized Hausdorff matrices.

Let m = 0 n = 0 a m n be an infinite double series with real or complex numbers, with partial sums

s m n = i = 0 m j = 0 n a i j .

For any double sequence ( u m n ) we shall define

Δ 11 u m n = u m n u m + 1 , n u m , n + 1 + u m + 1 , n + 1 .

Denote by A k 2 the sequence space defined by

A k 2 = { ( s m n ) m , n = 0 : m = 1 n = 1 ( m n ) k 1 | a m n | k < ; a m n = Δ 11 s m 1 , n 1 }

for k1.

A four-dimensional matrix T=( t m n i j :m,n,i,j=0,1,) is said to be absolutely k th power conservative for k1, if TB( A k 2 ); i.e., if

m = 1 n = 1 ( m n ) k 1 | Δ 11 s m 1 , n 1 | k <,

then

m = 1 n = 1 ( m n ) k 1 | Δ 11 t m 1 , n 1 | k <,

where

t m n = i = 0 j = 0 t m n i j s i j (m,n=0,1,),

see e.g. [9, 10] and the references contained therein.

A double Hausdorff matrix has entries

h m n i j = ( m i ) ( n j ) Δ 1 m i Δ 2 n j μ i j ,

where { μ i j } is any real or complex sequence and

Δ 1 m i Δ 2 n j μ i j = s = 0 m i t = 0 n j ( 1 ) i + j ( m i s ) ( n j t ) μ i + s j + t .

For double Hausdorff matrices, the necessary and sufficient condition for H to be conservative is the existence of a function χ(s,t)BV[0,1]×[0,1] such that

0 1 0 1 | d χ ( s , t ) | <,

and

μ m n = 0 1 0 1 s m t n dχ(s,t).

Quite recently, Savaş and Rhoades [8] extended the result of Das [7] to double Hausdorff summability. Their theorem is as follows.

Theorem 1 [8]

Let H be a conservative double Hausdorff matrix. Then HB( A k 2 ).

Our purpose is to achieve the result established in [7] for double E-J Hausdorff summability.

Main results

The matrix δ ( α , β ) =( δ m n i j ( α , β ) ), whose elements are defined by

δ m n i j ( α , β ) = { ( 1 ) i + j ( m + α m i ) ( n + β n j ) , i m , j n , 0 , otherwise ,

is called a difference matrix, where α and β are real numbers.

Theorem 2 The difference matrix δ ( α , β ) =( δ m n i j ( α , β ) ) is its own inverse.

Proof Let

a m n k l = i = 0 m j = 0 n δ m n i j ( α , β ) δ i j k l ( α , β ) ,

thus A= δ ( α , β ) δ ( α , β ) . For any double sequence ( u r s )

r = 0 m s = 0 n a m n r s u r s = r = 0 m s = 0 n i = 0 m j = 0 n δ m n i j ( α , β ) δ i j r s ( α , β ) u r s = r = 0 m s = 0 n ( 1 ) r + s u r s i = r m j = s n ( 1 ) i + j ( m + α m i ) ( n + β n j ) ( i + α i r ) ( j + β j s ) = r = 0 m s = 0 n ( 1 ) r + s u r s ( m + α m r ) ( n + β n s ) i = r m j = s n ( 1 ) i + j ( m r m i ) ( n s n j ) = u r s ,

since

i = r m j = s n ( 1 ) i + j ( m r m i ) ( n s n j ) = { ( 1 ) r + s , m = r , n = s , 0 , otherwise .

 □

Let ( μ m n ( α , β ) ) be a given sequence and μ ( α , β ) =( μ m n i j ( α , β ) ) be a diagonal matrix whose only non-zero entries are μ m n ( α , β ) = μ m n m n ( α , β ) . The transformation matrix

H ( α , β ) = δ ( α , β ) μ ( α , β ) δ ( α , β )

is called a double E-J generalized Hausdorff matrix corresponding to the sequence ( μ m n ( α , β ) ).

Theorem 3 A matrix H ( α , β ) =( h m n i j ( α , β ) ) is a double E-J generalized Hausdorff matrix corresponding to the sequence ( μ m n ( α , β ) ) if and only if its elements have the form

h m n i j ( α , β ) = ( m + α m i ) ( n + β n j ) Δ 1 m i Δ 2 n j μ i j ( α , β ) ,

where

Δ 1 m i Δ 2 n j μ i j ( α , β ) := r = 0 m i s = 0 n j ( 1 ) r + s ( m i r ) ( n j s ) μ i + r , j + s ( α , β ) .

Proof Let H ( α , β ) = δ ( α , β ) μ ( α , β ) δ ( α , β ) be a double E-J Hausdorff matrix. Applying this to a double sequence ( s m n ) we have

t m n = i = 0 m j = 0 n h m n i j ( α , β ) s i j = i = 0 m j = 0 n r = 0 m s = 0 n δ m n r s ( α , β ) μ r s ( α , β ) δ r s i j ( α , β ) s i j = i = 0 m j = 0 n r = 0 m s = 0 n ( 1 ) r + s ( m + α m r ) ( n + β n s ) μ r s ( α , β ) ( 1 ) i + j ( r + α r i ) ( s + β s j ) s i j = i = 0 m j = 0 n ( 1 ) i + j ( m + α m i ) ( n + β n j ) r = i m s = j n ( 1 ) r + s ( m i m r ) ( n j n s ) μ r s ( α , β ) s i j = i = 0 m j = 0 n ( m + α m i ) ( n + β n j ) r = 0 m i s = 0 n j ( 1 ) r + s ( m i r ) ( n j s ) μ i + r , j + s ( α , β ) s i j .

Hence

h m n i j ( α , β ) = ( m + α m i ) ( n + β n j ) r = 0 m i s = 0 n j ( 1 ) r + s ( m i r ) ( n j s ) μ i + r , j + s ( α , β ) .

 □

For double E-J Hausdorff matrices, the necessary and sufficient condition for H ( α , β ) to be conservative is the existence of a function χ(s,t)BV[0,1]×[0,1] such that

0 1 0 1 | d χ ( s , t ) | <,

and

μ m n ( α , β ) = 0 1 0 1 s m + α t n + β dχ(s,t).

Theorem 4 Given a function χ(s,t)BV[0,1]×[0,1], a bounded variation in the unit square, the corresponding double E-J Hausdorff transformation ( t m n ), of a sequence ( s m n ), may be defined by

t m n = i = 0 m j = 0 n ( m + α m i ) ( n + β n j ) s i j 0 1 0 1 s i + α ( 1 s ) m i t j + β ( 1 t ) n j dχ(s,t).

Proof For im and jn,

h m n i j ( α , β ) = k = i m l = j n δ m n k l ( α , β ) μ k l ( α , β ) δ k l i j ( α , β ) = k = i m l = j n δ m n k l ( α , β ) 0 1 0 1 s k + α t l + β d χ ( s , t ) δ k l i j ( α , β ) = 0 1 0 1 k = i m l = j n ( 1 ) k + l ( m + α m k ) ( n + β n l ) ( 1 ) i + j ( k + α k i ) ( l + β l j ) s k + α t l + β d χ ( s , t ) = 0 1 0 1 k = i m l = j n ( 1 ) k + l + i + j ( m + α m i ) ( n + β n j ) ( m i m k ) ( n j n l ) s k + α t l + β d χ ( s , t ) = 0 1 0 1 ( m + α m i ) ( n + β n j ) k = 0 m i l = 0 n j ( 1 ) k + l ( m i k ) ( n j l ) s k + i + α t l + j + β d χ ( s , t ) = 0 1 0 1 ( m + α m i ) ( n + β n j ) s i + α t j + β ( k = 0 m i l = 0 n j ( 1 ) k + l ( m i k ) ( n j l ) s k t l ) d χ ( s , t ) = 0 1 0 1 ( m + α m i ) ( n + β n j ) s i + α t j + β ( 1 s ) m i ( 1 t ) n j d χ ( s , t ) .

 □

Theorem 5 Let H ( α , β ) be a conservative double E-J Hausdorff matrix. Then H ( α , β ) B( A k 2 ), α,β0.

As tools to prove our result, we need to the following lemmas.

Lemma 1 [6]

Let k1, nv and α0. Then

E m + α k 1 E m μ μ + α 1 E μ + α k 1 E m μ μ + α + k 2 .

The following lemma is a double version of [11].

Lemma 2 For 0s1, 0t1, α0 and β0

i = 0 m j = 0 n ( m + α i ) ( n + β j ) ( 1 s ) m ( 1 t ) n s m + α i t n + β j 1.

Proof of Theorem 5 Let ( t m n ) be the double E-J transform of a double sequence ( s m n ); i.e.,

t m n = μ = 0 m v = 0 n h m n μ v ( α , β ) s μ v .

We will demonstrate that

m = 1 n = 1 ( m n ) k 1 | a m n | k < m = 1 n = 1 ( m n ) k 1 | Δ 11 t m 1 , n 1 | k <.
(1)

Write

t m n = μ = 0 m v = 0 n b μ v .

Then b m n = Δ 11 t m 1 , n 1 . For k1

E m k 1 = ( m + k 1 m ) = ( m + k 1 k 1 ) = ( m + k 1 ) ! m ! ( k 1 ) ! = Γ ( m + k ) Γ ( m + 1 ) Γ ( k ) .

Then

E m k 1 m k 1 Γ ( k ) m k 1 , m k 1 E m k 1 E m + α k 1 .

Due to this (1) is equivalent to

m = 1 n = 1 E m + α k 1 E n + β k 1 | a m n | k < m = 1 n = 1 E m + α k 1 E n + β k 1 | b m n | k <.
(2)

For s[0,1] and t[0,1] define

ϕ m n (s,t)= μ = 1 m v = 1 n E m μ μ + α 1 E n v v + β 1 s μ + α t v + β ( 1 s ) m μ ( 1 t ) n v a μ v .
(3)

It follows from the Hölder inequality that

| ϕ m n ( s , t ) | k = | μ = 1 m v = 1 n E m μ μ + α 1 E n v v + β 1 s μ + α t v + β ( 1 s ) m μ ( 1 t ) n v a μ v | k μ = 1 m v = 1 n E m μ μ + α 1 E n v v + β 1 s μ + α t v + β ( 1 s ) m μ ( 1 t ) n v | a μ v | k × { μ = 1 m v = 1 n E m μ μ + α 1 E n v v + β 1 s μ + α t v + β ( 1 s ) m μ ( 1 t ) n v } k 1 .

From Lemma 2

μ = 1 m v = 1 n E m μ μ + α 1 E n v v + β 1 s μ + α t v + β ( 1 s ) m μ ( 1 t ) n v = μ = 1 m v = 1 n ( m + α 1 m μ ) ( n + β 1 n v ) s μ + α t v + β ( 1 s ) m μ ( 1 t ) n v = μ = 0 m 1 v = 0 n 1 ( m + α 1 m μ 1 ) ( n + β 1 n v 1 ) s μ + α + 1 t v + β + 1 ( 1 s ) m μ 1 ( 1 t ) n v 1 = s t μ = 0 m 1 v = 0 n 1 ( m + α 1 m μ 1 ) ( n + β 1 n v 1 ) s μ + α t v + β ( 1 s ) m μ 1 ( 1 t ) n v 1 = O ( s t ) .

Hence

| ϕ m n (s,t) | k =O(1) ( s t ) k 1 μ = 1 m v = 1 n E m μ μ + α 1 E n v v + β 1 s μ + α t v + β ( 1 s ) m μ ( 1 t ) n v | a μ v | k

and from Lemma 1

m = 1 n = 1 E m + α k 1 E n + β k 1 | ϕ m n ( s , t ) | k = O ( 1 ) m = 1 n = 1 E m + α k 1 E n + β k 1 ( s t ) k 1 × μ = 1 m v = 1 n E m μ μ + α 1 E n v v + β 1 s μ + α t v + β ( 1 s ) m μ ( 1 t ) n v | a μ v | k = O ( 1 ) ( s t ) k 1 μ = 1 v = 1 s μ + α t v + β | a μ v | k × m = μ n = v E m + α k 1 E n + β k 1 E m μ μ + α 1 E n v v + β 1 ( 1 s ) m μ ( 1 t ) n v = O ( 1 ) ( s t ) k 1 μ = 1 v = 1 s μ + α t v + β | a μ v | k E μ + α k 1 E v + β k 1 × m = μ n = v E m μ μ + α + k 2 E n v v + β + k 2 ( 1 s ) m μ ( 1 t ) n v = O ( 1 ) ( s t ) k 1 μ = 1 v = 1 s μ + α t v + β | a μ v | k E μ + α k 1 E v + β k 1 s μ α k + 1 t v β k + 1 = O ( 1 ) μ = 1 v = 1 E μ + α k 1 E v + β k 1 | a μ v | k .

From Lemma of [8], if ( t m n ) and ( τ m n ) are the (H, μ m n ) transformation of ( s m n ) and (mn a m n ), respectively, then

τ m n =mn Δ 11 t m 1 , n 1 .

A similar consequence can be proved for ( H ( α , β ) , μ ( α , β ) ), see [6]; i.e.,

τ m n =(m+α)(n+β) Δ 11 t m 1 , n 1 .

Hence

b m n = 1 ( m + α ) ( n + β ) τ m n = 1 ( m + α ) ( n + β ) i = 0 m j = 0 n ( m + α m i ) ( n + β n j ) Δ 1 m i Δ 2 n j μ i j ( α , β ) ( i + α ) ( j + β ) a i j = i = 0 m j = 0 n ( m + α 1 m i ) ( n + β 1 n j ) Δ 1 m i Δ 2 n j μ i j ( α , β ) a i j = i = 0 m j = 0 n E m i i + α 1 E n j j + β 1 Δ 1 m i Δ 2 n j μ i j ( α , β ) a i j .

Since H ( α , β ) is conservative, μ n ( α , β ) is a moment sequence,

μ m n ( α , β ) = 0 1 0 1 s m + α t n + β dχ(s,t),

and

Δ 1 m i Δ 2 n j μ i j ( α , β ) = 0 1 0 1 s i + α ( 1 s ) m i t j + β ( 1 t ) n j dχ(s,t)

from Theorem 4. In view of (3) we can deduce that

b m n = i = 0 m j = 0 n E m i i + α 1 E n j j + β 1 0 1 0 1 s i + α ( 1 s ) m i t j + β ( 1 t ) n j d χ ( s , t ) a i j = 0 1 0 1 ( i = 0 m j = 0 n E m i i + α 1 E n j j + β 1 s i + α t j + β ( 1 s ) m i ( 1 t ) n j a i j ) d χ ( s , t ) = 0 1 0 1 ϕ m n ( s , t ) d χ ( s , t ) .

Using Minkowski’s inequality we get

{ m = 1 n = 1 E m + α k 1 E n + β k 1 | b m n | k } 1 / k = { m = 1 n = 1 E m + α k 1 E n + β k 1 | 0 1 0 1 ϕ m n ( s , t ) d χ ( s , t ) | k } 1 / k 0 1 0 1 | d χ ( s , t ) | { m = 1 n = 1 E m + α k 1 E n + β k 1 | ϕ m n ( s , t ) | k } 1 / k = O ( 1 ) 0 1 0 1 | d χ ( s , t ) | { μ = 1 v = 1 E μ + α k 1 E v + β k 1 | a μ v | k } 1 / k .

Therefore the proof of Theorem 5 is complete. □

Specially, if we take α=0 and β=0 in Theorem 5, we get Theorem 1 as a corollary.

The following is an example of a double E-J Hausdorff matrix.

A doubly infinite Cesàro matrix (C,γ,δ) is a doubly infinite Hausdorff matrix with entries

h m n i j = ( m + γ i 1 n i ) ( n + δ j 1 n j ) ( m + γ γ ) ( n + δ δ ) ,γ,δ0.

We use the following to denote the corresponding E-J generalizations of the (C,γ,δ).

( C ( α , β ) ,γ,δ) has moment sequence

μ m n ( α , β ) = 0 1 0 1 u m + α v n + β γδ ( 1 u ) γ 1 ( 1 v ) δ 1 dudv,

where

χ(u,v)=γδ 0 u 0 v ( 1 s ) γ 1 ( 1 t ) δ 1 dsdt.

For im and jn,

h m n i j ( α , β ) = 0 1 0 1 ( m + α m i ) ( n + β n j ) u i + α v j + β ( 1 u ) m i ( 1 v ) n j d χ ( u , v ) = 0 1 0 1 ( m + α m i ) ( n + β n j ) × u i + α v j + β ( 1 u ) m i ( 1 v ) n j γ δ ( 1 u ) γ 1 ( 1 v ) δ 1 d u d v = γ δ ( m + α m i ) ( n + β n j ) 0 1 0 1 u i + α ( 1 u ) m i + γ 1 v j + β ( 1 v ) n j + δ 1 d u d v = γ δ ( m + α m i ) ( n + β n j ) B ( i + α + 1 , m i + γ ) B ( j + β + 1 , n j + δ ) = γ Γ ( m + α + 1 ) Γ ( m i + γ ) Γ ( m i + 1 ) Γ ( m + α + γ + 1 ) δ Γ ( n + β + 1 ) Γ ( n j + δ ) Γ ( n j + 1 ) Γ ( n + β + δ + 1 ) = E m i γ 1 E n j δ 1 E m + α γ E n + β δ .

For the special case γ,δ=1,

( C ( α , β ) , 1 , 1 ) = { 1 ( m + α + 1 ) ( n + β + 1 ) , i m  and  j n , 0 , otherwise

is a double E-J Hausdorff matrix.

References

  1. 1.

    Adams CR: Hausdorff transformations for double sequences. Bull. Am. Math. Soc. 1933, 39: 303–312. 10.1090/S0002-9904-1933-05621-5

  2. 2.

    Ramanujan MS: On Hausdorff transformations for double sequences. Proc. Indian Acad. Sci. Sect. A. 1955, 42: 131–135.

  3. 3.

    Ustina F: The Hausdorff means for double sequences. Can. Math. Bull. 1967, 10: 347–352. 10.4153/CMB-1967-031-1

  4. 4.

    Endl K: Untersuchungen über momentenprobleme bei verfahren vom Hausdorffschen typus. Math. Ann. 1960, 139: 403–432. 10.1007/BF01342846

  5. 5.

    Jakimovski, A: The product of summability methods; part 2. Technical Report 8, Jerusalem (1959)

  6. 6.

    Savaş E, Şevli H: Generalized Hausdorff matrices as bounded operators over A k . J. Comput. Anal. Appl. 2009, 11: 702–710.

  7. 7.

    Das G: A Tauberian theorem for absolute summability. Proc. Camb. Philos. Soc. 1970, 67: 321–326. 10.1017/S0305004100045606

  8. 8.

    Savaş E, Rhoades BE: Every conservative double Hausdorff matrix is a k -th absolutely summable operator. Anal. Math. 2009, 35: 249–256. 10.1007/s10476-009-0401-0

  9. 9.

    Savaş E, Şevli H: On absolute summability for double triangle matrices. Math. Slovaca 2010, 60: 495–506. 10.2478/s12175-010-0028-4

  10. 10.

    Şevli H, Savaş E: Some further extensions of absolute Cesàro summability for double series. J. Inequal. Appl. 2013., 2013: Article ID 144

  11. 11.

    Jakimovski A, Ramanujan MS: A uniform approximation theorem and its application to moment problems. Math. Z. 1964, 84: 143–153. 10.1007/BF01117122

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Acknowledgements

This work is supported by Istanbul Commerce University Scientific Research Projects Coordination Unit.

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Correspondence to Hamdullah Şevli.

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The authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.

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Keywords

  • absolute summability
  • conservative matrix
  • double series
  • Hausdorff matrices