Abstract ad Applied Aalysis Volume 2011, Article ID 527360, 5 pages doi:10.1155/2011/527360 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, 34349 Istabul, Turkey 2 Departmet of Mathematics, Istabul Commerce Uiversity, Uskudar, 34672 Istabul, Turkey 3 Departmet of Mathematics, Idiaa Uiversity, Bloomigto, IN 47405-7106, 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,... 1.1 0 0 always imply α 0 0, 1, 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.
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, 2... 1.4 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 1. 1.5 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, 2... 1.6 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 1. 1.7 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 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 1.10. 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. 40 50, 1930. 2 J. D. Hill, O perfect methods of summability, Duke Mathematical Joural, vol. 3, o. 4, pp. 702 714, 1937. 3 B. E. Rhoades, Some Hausdorff matrices ot of type M, Proceedigs of the America Mathematical Society, vol. 15, pp. 361 365, 1964. 4 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, 1917. 5 F. Hausdorff, Summatiosmethode ud Mometfolge. I, Mathematische Zeitschrift, vol. 9, o. 1-2, pp. 74 109, 1921. 6 K. Edl, Abstracts of short commuicatios ad scietific program, i Proceedigs of the Iteratioal Cogress of Mathematicias, vol. 73, p. 46, 1958. 7 K. Edl, Utersuchuge über Mometeprobleme bei Verfahre vom Hausdorffsche Typus, Mathematische Aale, vol. 139, pp. 403 432, 1960. 8 A. Jakimovski, The product of summability methods; New classes of trasformatios ad their properties, Tech. Rep. AF61, 1959, 052-187.
Abstract ad Applied Aalysis 5 9 G. H. Hardy, Diverget Series, Claredo Press, Oxford, UK, 1949. 10 A. Wilasky, A applicatio of Baach liear fuctioals to summability, Trasactios of the America Mathematical Society, vol. 67, pp. 59 68, 1949. 11 A. Wilasky ad K. Zeller, Summatio of bouded diverget sequeces, topological methods, Trasactios of the America Mathematical Society, vol. 78, pp. 501 509, 1955. 12 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 131240, 14 pages, 2011.
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