Research Article Some E-J Generalized Hausdorff Matrices Not of Type M

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1 Abstract ad Applied Aalysis Volume 2011, Article ID , 5 pages doi: /2011/ Research Article Some E-J Geeralized Hausdorff Matrices Not of Type M T. Selmaogullari, 1 E. Savaş, 2 ad B. E. Rhoades 3 1 Departmet of Mathematics, Mimar Sia Fie Arts Uiversity, Besiktas, Istabul, Turkey 2 Departmet of Mathematics, Istabul Commerce Uiversity, Uskudar, Istabul, Turkey 3 Departmet of Mathematics, Idiaa Uiversity, Bloomigto, IN , USA Correspodece should be addressed to T. Selmaogullari, tugcemat@yahoo.com Received 23 April 2011; Accepted 4 August 2011 Academic Editor: Alberto d Oofrio Copyright q 2011 T. Selmaogullari et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We show that there exists a regular E-J geeralized Hausdorff matrix which has o zero elemets o the mai diagoal ad which is ot of type M ad establish several other related theorems. 1. Itroductio The covergece domai of a ifiite matrix A a k, k 0, 1,... will be deoted by A ad is defied by A : {x {x } A x c}, where c deotes the space of covergece sequeces ad A x : k0 a kx k. If for two matrices A ad B, we have the relatio A B, we say that B is ot weaker tha A. The ecessary ad sufficiet coditios of Silverma ad Toeplitz for a matrix to be coservative some authors use the word covergece-preservig istead of coservative are as lim a k a k exists for each k, lim k0 a k t exists, follows: A : sup k0 a k <. A coservative matrix A is called multiplicative if each a k 0 ad regular if, i additio, t 1. If A is a coservative matrix, the χa lim k a k k lim a k is called the characteristic of A. A coservative matrix A is called coregular if χa / 0 ad coull if χa 0. Regular matrices are coregular, sice χa 1. A matrix A a k, k 0, 1,... is called triagular if a k 0 for all k>, ad it is called a ormal if it is, triagular ad a / 0 for all. Let A a k, k 0, 1,... deote a ifiite matrix. The A is said to be of type M if the coditios α <, α a k 0 k 0, 1, 2, always imply α 0 0, 1, 2,...

2 2 Abstract ad Applied Aalysis Matrices of type M were first itroduced by Mazur 1 ad amed by Hill 2. Hill 2 developed several sufficiet coditios for a Hausdorff matrix to be of Type M. He showed that there exists a regular Hausdorff matrix which has a zero o the mai diagoal, ot of type M. He also posed the followig questio: does there exist a regular Hausdorff matrix which has o zero elemets o the mai diagoal ad which is ot of type M? Rhoades 3 aswered the above questio i the affirmative ad established several other related theorems. I this paper, we aswer the above questio for E-J geeralized Hausdorff matrices. We use the words fiite sequece to describe a sequece which is cotaiig oly a fiite umber of ozero terms. It is clear that a triagular matrix which is ot a ormal caot be of type M, sice a fiite sequece ca be foud satisfyig 1.1. Also, if a matrix is a ormal, there ca be o fiite sequece as a solutio of 1.1. All diagoal matrices with ozero diagoal elemets are of type M. Hausdorff matrices were show by Hurwitz ad Silverma 4 to be the class of triagular matrices that commute with C, thecesáro matrix of order oe. Hausdorff 5 reexamied this class, i the process of solvig the momet problem over a fiite iterval, ad developed may of the properties of the matrices that ow bear his ame. Several geeralizatios of Hausdorff matrices have bee made. I this paper we will be cocered with the geeralized Hausdorff matrices as defied idepedetly by Edl 6, 7 ad Jakimovski 8. A geeralized Hausdorff matrix H α is a lower triagular ifiite matrix with etries ( ) α h α k Δ k k, 0 k, 1.2 k where α is real umber, is a real sequece, ad Δ is forward differece operator defied by Δ k k k1, Δ 1 k ΔΔ k. We will cosider here oly oegative α. For α 0 oe obtais a ordiary Hausdorff matrix. From 6 or 8, a geeralized Hausdorff matrix for α>0 is regular if ad oly if there exists a fuctio χ BV 0, 1 with χ1 χ0 1 such that α 1 0 t α dχt, 1.3 i which case α is called the momet sequece for H α ad χ is called the momet geeratig fuctio, or mass fuctio, for H α. For ordiary Hausdorff summability see, e.g., 9, the ecessary ad sufficiet coditios for regularity are that the fuctio χ BV 0, 1, χ1 χ0 1,χ0 χ0 0, ad 1.3 is satisfied with α 0. The purpose of this paper is to show that there exists a regular E-J geeralized Hausdorff matrix which has o zero elemets o the mai diagoal ad which is ot of type M ad establish several other related theorems. The followig is a cosequece of 10, Theorem 3.2.1d. Theorem 1.1 see 10. If A is a ormal, coservative ad coregular, the A is of type M if ad oly if c A. The followig is a cosequece of 11, Theorem 1a ad c.

3 Abstract ad Applied Aalysis 3 Theorem 1.2 see 11. Let A be coservative. c is closed i A if ad oly if A sums o bouded diverget sequeces. Theorem 1.3. Let α b a a>0, b > 0, α 0 0, 1, a α b α The the correspodig regular E-J geeralized Hausdorff matrix is ot of type M. Proof. If a is a positive iteger, the H α is ot of type M as remarked above, sice it has a zero o its diagoal. Assume a is ot a positive iteger. From 12, the covergece domais for E-J geeralized Hausdorff matrices with momet geeratig sequece as defied above are H α c x α, where x α Γ α 1 Γ a Sice H α sums o bouded diverget sequeces, from Theorem 1.2, c is closed i H α. From Theorem 1.1,sicec is ot dese i the covergece domai of each H α, oe of them is of type M. Let A be a coservative matrix. If c is dese i A the A is called perfect. For certai classes of matrices, perfectess ad type M are closely related. Note that, from Theorem 1.1, type M ad perfectess are equivalet for ormal, coservative, ad coregular matrices. If oe examies the sequeces of Theorem 1.3 for a a iteger, the oe otes that the correspodig matrices are ot ormal. It remais to determie if each such matrix is perfect. Theorem 1.4. Let α b r, r is a positive iteger, b>0, α 0 0, 1, r α b α The the correspodig regular E-J geeralized Hausdorff matrix is ot perfect. Proof. I 12, it is proved that H α H α c x α, where is a regular E-J geeralized Hausdorff matrix with x α Γ α 1 Γ r As metioed i the proof of Theorem 1.3, c is ot dese i the covergece domai of each H α. Hece each H α is ot perfect. Theorem 1.5 see 2. The product AB C of two triagular perfect methods A ad B is also a triagular perfect method.

4 4 Abstract ad Applied Aalysis Theorem 1.6 see 2. If the product AB C of two triagular covergece-preservig methods A ad B is of type M,theA must be of type M. Theorem 1.7 see 2. If A is ormal ad B is triagular, the B is ot weaker tha A if ad oly if BA 1 is covergece preservig. Theorem 1.8. If H α H α,theh α is ot of type M. is ormal, coservative, ad ot of type M, ad if H α is ot weaker tha Proof. By Theorem 1.7, H α ϕ H α Hα 1 is coservative. From the defiitio of E-J geeralized Hausdorff matrix, it is easily show that multiplicatio of a E-J geeralized Hausdorff matrix is commutative ad the result is agai a E-J geeralized Hausdorff matrix. Also the iverse of a ormal E-J geeralized Hausdorff matrix is also a ormal E-J geeralized Hausdorff matrix. Thus H α ϕ oce from Theorem 1.6. Theorem 1.9. If H α the H α is of type M. H α 1 H α ad H α is ormal ad coservative ad if H α Proof. As i the previous theorem we have H α ϕ coservative. The H α H α Hα H α H α ϕ, ad the result follows at is of type M ad ot weaker tha H α, H α H α 1 H α 1 H α, where H α ϕ is ϕ, ad the coclusio follows agai from Theorem 1.6. From Theorem 1.5 ad the multiplicatio facts of E-J geeralized Hausdorff matrix, we obtai the followig theorem. Theorem The product of a fiite umber of perfect E-J geeralized Hausdorff methods is likewise a perfect E-J geeralized Hausdorff method. Ackowledgmet The first author ackowledges support from the Scietific ad Techical Research Coucil of Turkey i the preparatio of this paper. Refereces 1 S. Mazur, Eie Awedug der Theorie der Operatioe bei der Utersuchug der Toeplitzche Limitierugsverfahre, Studia Mathematica, vol. 2, pp , J. D. Hill, O perfect methods of summability, Duke Mathematical Joural, vol. 3, o. 4, pp , B. E. Rhoades, Some Hausdorff matrices ot of type M, Proceedigs of the America Mathematical Society, vol. 15, pp , W. A. Hurwitz ad L. L. Silverma, O the cosistecy ad equivalece of certai defiitios of summability, Trasactios of the America Mathematical Society, vol. 18, o. 1, pp. 1 20, F. Hausdorff, Summatiosmethode ud Mometfolge. I, Mathematische Zeitschrift, vol. 9, o. 1-2, pp , K. Edl, Abstracts of short commuicatios ad scietific program, i Proceedigs of the Iteratioal Cogress of Mathematicias, vol. 73, p. 46, K. Edl, Utersuchuge über Mometeprobleme bei Verfahre vom Hausdorffsche Typus, Mathematische Aale, vol. 139, pp , A. Jakimovski, The product of summability methods; New classes of trasformatios ad their properties, Tech. Rep. AF61, 1959,

5 Abstract ad Applied Aalysis 5 9 G. H. Hardy, Diverget Series, Claredo Press, Oxford, UK, A. Wilasky, A applicatio of Baach liear fuctioals to summability, Trasactios of the America Mathematical Society, vol. 67, pp , A. Wilasky ad K. Zeller, Summatio of bouded diverget sequeces, topological methods, Trasactios of the America Mathematical Society, vol. 78, pp , T. Selmaogullari, E. Savaş, ad B. E. Rhoades, Size of covergece domais for geeralized Hausdorff prime matrices, Joural of Iequalities ad Applicatios, vol. 2011, Article ID , 14 pages, 2011.

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