Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 131240, 14 pages doi:10.1155/2011/131240 Research Article Size of Convergence Domains for Generalized Hausdorff Prime Matrices T Selmanogullari,1 E Savas,2 and B E Rhoades3 ¸ Department of Mathematics, Mimar Sinan Fine Arts University, Besiktas, 34349 Istanbul, Turkey Department of Mathematics, Istanbul Commerce University, Uskudar, 34672 Istanbul, Turkey Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA Correspondence should be addressed to T Selmanogullari, tugcenmat@yahoo.com Received December 2010; Accepted March 2011 Academic Editor: Q Lan Copyright q 2011 T Selmanogullari et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We show that there exit E-J generalized Hausdorff matrices and unbounded sequences x such that each matrix has convergence domain c ⊕ x Introduction The convergence domain of an infinite matrix A ank n, k 0, 1, will be denoted by A and is defined by A : {x {xn } | An x ∈ c}, where c denotes the space of convergence sequences, An x : k ank xk The necessary and sufficient conditions of Silverman and Toeplitz for a matrix to be conservative are limn ank ak exists for each k, limn ∞ ank t k exists, and ||A|| : supn ∞ |ank | < ∞ A conservative matrix A is called multiplicative if k each ak and regular if, in addition, t The E-J generalized Hausdorff matrices under consideration were defined indepenα dently by Endl 1, and Jakimovski Each matrix Hμ is a lower triangular matrix with nonzero entries ⎛ ⎞ n α α ⎠Δn−k μk , 1.1 hnk ⎝ n−k where α is real number, {μn } is a real or complex sequence and Δ is forward difference μk − μk , Δn μk Δ Δn μk We will consider here only operator defined by Δμk nonnegative α For α 0, one obtains an ordinary Hausdorff matrix 2 Journal of Inequalities and Applications From or a E-J generalized Hausdorff matrix for α > is regular if and only if there exists a function χ ∈ BV 0, with χ − χ such that α μn tn α dχ t , 1.2 α α in which case χ is called the moment generating function, or mass function, for Hμ and μn is called moment sequence For ordinary Hausdorff summability , the necessary and sufficient conditions, for regularity are that function χ ∈ BV 0, , χ − χ 1, χ χ , and 1.2 is satisfied with α As noted in , the set of all multiplicative Hausdorff matrices forms a commutative Banach algebra that is also an integral domain, making it possible to define the concepts of unit, prime, divisibility, associate, multiple, and factor Hille and Tamarkin 6, , using some techniques from , showed that every Hausdorff matrix with moment function z−a , z b μz R a > 0, R b >0 1.3 is prime In 1967, Rhoades showed that the convergence domain of every known prime Hausdorff matrix is of the form c ⊕ x for a particular unbounded sequence x Given any unbounded sequence x, Zeller 10 constructed a regular matrix A with convergence domain A c ⊕ x It has been shown by Parameswaran 11 that if x is any unbounded sequence such that {xn − xn−1 } is bounded, divergent, and Borel summable, then no Hausdorff matrix H exists with H c ⊕ x The main result of this paper is to show that there exist E-J generalized Hausdorff α matrices Hμ whose moment sequences are α μn n n−a , b α R a > 0, R b > 0, 1.4 and unbounded sequences x α such that each matrix has convergent domain c ⊕ x α Define the sequences x α by Γn α Γ n−a α xn 1 for R a > 0, 1.5 α where it is understood that if a is positive integer, then xn for n 0, 1, , a − α If λn is the moment sequence defined by n − a / n α , R a > 0, then it is clear α α that Hμ Hλ Hence, it will be sufficient to prove the theorem by using b 1, in 1.4 To have the convenience of regularity, we will use the sequence α μn since the constant −1/ a − a n−a α n α , 1.6 α α does not affect the size of the convergence domain of Hμ Journal of Inequalities and Applications Auxiliary Results In order to prove the main theorem of this paper, we will need the following results Lemma 2.1 Let A, B ∈ C, d, n ∈ N ∪ {0}, d ≤ n Then, formally, for any n, n k ΓA k ΓB k d A−B ΓA n Γ A d − ΓB n Γ B−1 d 2.1 Proof Lemma 2.1 appears as formula 12 on page 138 of 12 α Lemma 2.2 For m, n integers n > m > a, xn as in 1.5 , n α k m xk 1 a k−a α − α xm α xn 2.2 Proof Using Lemma 2.1, n α k m xk n k−a Γ k−a Γk α 1 k m Γ n Γn −a − α 1 a α xm α − α xn 1−a Γ m − α Γm 1−a α 2.3 Lemma 2.3 For ≤ r ≤ a − n−1 j a α hnj −1 α hjr α xn Γa α − a α n−a 2.4 α Proof μn can be written as α μn −1 a α a a α α n α , 2.5 Journal of Inequalities and Applications α so that, for ≤ k < n, hnk − n−1 a −1 α hnj α α 1/ a − hjr α n α n−1 −xn a j a xj α a j a 1 xn a α From Lemma 2.1 and 3.11 , α a j −a a α a α xn Γa α − Γ a a α Γ j −a j a 1Γ xn α α α α j n−1 α α j α α − 2.6 Γ n−a Γn α α n−a Main Result α Theorem 3.1 If for fixed a and b the matrix Hμ is defined by 1.4 and a sequence x α by 1.5 , α then Hμ c⊕x α α Proof We will first show that c ⊕ x α ⊆ Hμ We can write the matrix α Hμ α −1/ a α I − Hλ a n α α α , where the diagonal entries of Hλ are α λn 3.1 For each n and k, α Δn−k λk a α tk α 1−t n−k dt a 3.2 Γ k α Γ n−k Γn α α Therefore, n α Hλ α n−k n,k n α Δn−k λk α a n−k a n α α α Γ k α Γ n−k Γn α 3.3 Journal of Inequalities and Applications Define yn −un / a α , where n α xn − un α k α hn,k xk 3.4 From Lemma 2.1, Γn α Γ n−a − n Γn α Γ n−a un 1−1 Γ α Γ −a n α a α α 1 α a Γn α Γ n−a Γ α Γ −a n α Γα − Γ −a 3.5 −→ as n −→ ∞ This argument is valid provided a is not a positive integer If a is a positive integer, then α xk for ≤ k ≤ a − Then, un for ≤ n ≤ a − 1, and for n ≥ a, from Lemma 2.1, we get un α xn − a n α α 1 n k Γ k α Γ k−a a 1 Γn α Γ n−a a − n Γn α Γ n−a Γa α − − n α n Γn α Γ n−a 1 1− α α n n Γa 1 1 α Γ 1 − α n Γn α α Γ n−a Γn α Γ n−a Γa α − Γ1 Γa α n α 3.6 α α −→ as n −→ ∞ α α α Since Hμ is regular, c ⊆ Hμ Thus, c ⊕ x α ⊆ Hμ To prove the converse, we will use Zeller’s technique to construct a regular matrix A α with A c ⊕ x α and then show that Hμ ⊆ A and define a sequence {Pn } inductively by selecting Pn to be smallest Set P0 α α integer P > Pn such that |xP | ≥ |xPn | Such a construction is clearly possible, since x α is α not bounded Let qn α α − xPn−1 /xPn , n b00 bn,n−1 1, 2, Define a matrix B by 1, α qn n ≥ 1, , 3.7 α bn,n − bn,k xPn−1 α α qn xPn , n ≥ 1, otherwise 6 Journal of Inequalities and Applications Now, define the matrix A as follows: bnk , aPn ,Pk aPn ,k ann 0, k / Pi for any integer i, 1, 3.8 n / Pi for any integer i If n / Pi for any integer i, then there exists an integer r such that Pr < n < Pr For this r, define α xn an,Pr−1 α α xPr − xPr−1 , 3.9 α −xn α α xPr − xPr−1 an,Pr Set ank otherwise From 10 , A is regular and A c ⊕ x α There are three cases to consider, based on whether a is real number and not a positive integer, a is positive integer, or a is complex Proof of Case I If a is real and not a positive integer, the E-J generalized Hausdorff matrix α α α hnk −1 with generating Hμ generated by 1.6 has a unique two sided inverse Hμ −1 sequence − a α μn α n n−a α −a α − a α a α n−a 3.10 For k < n, −1 α hnk n α n−k − a α −xn α α To show that Hμ Each column of −1 − a α μk α a α Γ n α Γ k−a Γ k α Γ n−a a α xk hnn Δn−k α a α k−a α n n−a α , ⊆ A , it will be sufficent to show that D α Hμ −1 3.11 3.12 α A Hμ −1 is a regular matrix is essentially a scalar multiple of 1.5 , so it is obvious that each Journal of Inequalities and Applications α −1 column of Hμ belongs to the convergence domain of A However, it will be necessary to calculate the terms of D explicitly, since we must show that t and that D has finite norm If k / Pi for any integer i, and r denotes the integer such that Pr−1 < k < Pr , then from the definition of A, n −1 α dPn ,k j r aPn ,Pj hPj ,k α bn,n−1 hPn−1 ,k −1 3.13 −1 α bnn hPn ,k If k Pr for r < n − 1, then n α dPn ,Pr j r For k aPn ,Pj hPj ,Pr −1 3.14 Pn−1 , dPn ,Pn−1 α aPn ,Pn−1 hPn−1 ,Pn−1 − a α qn − a α qn α −1 α aPn ,Pn hPn ,Pn−1 ⎛ α Pn−1 α Pn−1 − a ⎝ −1 α −xPn−1 α qn α xPn ⎞⎛ ⎠⎝ − a α a α xPn−1 α α xPn Pn−1 − a ⎞ ⎠ 3.15 For Pn−1 < k < Pn , dPn ,k dPn ,Pn aPn ,Pn −1 α hPn ,k a −1 a α aPn ,Pn hPn ,Pn α α a α xPn−1 α α k − a qn xk α 3.16 , α Pn xPn−1 α α α Pn − a qn xPn 3.17 For n / Pi for any i, if we now let r denote the integer such that Pr < n < Pr , then for < k < Pr−1, n dnk j k α an,j hj,k −1 3.18 α an,Pr−1 hPr−1 ,k −1 α an,Pr hPr ,k −1 α an,n hn,k −1 8 Journal of Inequalities and Applications For k Pr−1 , −1 α an,Pr−1 hPr−1 ,Pr−1 dn,Pr−1 α xn − a α Pr−1 − a α xn − a α Pr−1 − a α an,Pr hPr ,Pr−1 −1 ⎛ −1 α an,n hn,Pr−1 ⎞ α xPr ⎝ Pr−1 α − a α α α α α α xPr − xPr−1 xPr − xPr−1 xPr−1 a α α xPr−1 1⎠ ⎞ ⎛ ⎝ Pr−1 α α α xPr − xPr−1 a − α α 3.19 ⎠ α xPr − xPr−1 α − a α xn α α xPr − xPr−1 For Pr−1 < k < Pr , a α a α −1 α an,n hn,k α α α 3.20 α xn xPr−1 α xPr − xPr−1 xk For k −1 α an,Pr hPr ,k dnk k−a Pr , dn,Pr −1 α an,Pr hPr ,Pr −1 α an,n hn,Pr ⎡ α − a α xn ⎣ − Pr α α α Pr − a xPr − xPr−1 ⎤ 1⎦ α α α xn α xPr α xPr − − Pr α xPr−1 3.21 xPr α −a Pr − a a α α α α α The quantity in brackets is equal to − Pr − a xPr − xPr−1 a dn,Pr α a α α α xPr − xPr−1 Pr − a xPr − xPr−1 α xPr−1 α xPr α a α α a xn a , giving α α xn α α xPr − xPr−1 xPr α 3.22 For Pr < k < n, dn,k α an,n hn,k −1 α −xn α xk a α a α k−a , 3.23 Journal of Inequalities and Applications and finally, − a dn,n α n n−a α 3.24 By using 3.13 – 3.17 , Pn −1 Pn dPn ,k dPn ,Pn−1 dPn ,k k dPn ,Pn k Pn−1 ⎡ a α ⎣ −1 α qn Pn −1 α xP n−1 a α Pn α k Pn−1 xk α Pn − k−a α xPn−1 α a xPn 3.25 ⎤ ⎦ By using Lemma 2.2, and noting that Pn α Pn − a 1 Pn a dPn ,k a α , Pn − a ⎡ α ⎣ −1 α qn k a ⎡ α ⎣ −1 α α xPn−1 a α a α a qn a α ⎞ ⎛ ⎝ 1 ⎠ − α α α xPn−1 xPn −1 − a α xPn−1 α a α α xPn −1 α α xPn−1 a α xPn−1 Pn − a x α α α α xPn−1 α xPn Pn ⎦ xPn Pn a α xPn−1 Pn − a x α ⎤ ⎤ ⎦ 3.26 Note that − a α xPn−1 α a α α xPn −1 ⎡ α a α xPn−1 Pn − a x α a α ⎣− α Pn ⎡ a a − a α Γ Pn α Γ Pn − a Γ Pn α a ⎦ α xPn−1 Γ Pn − a xPn−1 Γ Pn − a Pn − a Γ Pn α ⎡ α Γ Pn − a ⎣ −xPn−1 Γ Pn α a α α a α Pn − a xPn α xPn −1 α ⎣− α xPn−1 α a ⎤ α xPn−1 α xPn−1 Pn α ⎤ α ⎦ ⎤ ⎦ α xPn−1 α 3.27 10 Journal of Inequalities and Applications Finally, ⎡ Pn α ⎣ −1 a ⎡ α ⎣ −1 α a a dPn ,k α qn k a α a a xPn a qn xPn−1 a α Γ Pn a qn a α α ⎛ ⎞ α xPn−1 α ⎝ 1− α ⎠ − α qn xPn a α xPn−1 α a α α ⎤ ⎦ ⎤ ⎦ xPn α xPn−1 α xPn α α ⎣ −1 α Γ Pn − a ⎞ ⎛ α xPn−1 ⎝ 1− α ⎠ α xPn ⎡ α α a − α α α qn α xPn−1 3.28 ⎤ ⎦ For n / Pi for any i, r the integer such that Pr < n < Pr , and using 3.18 – 3.24 , we have n ⎡ α α xn ⎣ −1 α − xPr−1 a dnk α xPr k α xPr−1 Pr −1 a α α α k Pr−1 xk xPr−1 a k−a α α xPr Pr − a ⎤ ⎦ 3.29 α −xn n−1 a α a α xk k Pr α − k−a a α α n n−a α n−a Writing n α / n − a xn a α /xn in brackets, which we call I1 , takes the form I1 α xPr−1 a α a ⎞ ⎛ ⎝ 1 ⎠ − α α α xPr−1 xPr −1 α and using Lemma 2.2, the quantity α xPr−1 a α α xPr Pr − a ⎞ ⎛ α xPr−1 a 1⎝ α a α α xPr−1 − a α α xPr −1 α xPr Pr − a 3.30 ⎠ The sum − a α α xPr −1 α xPr Pr − a − Γ Pr − a a α Γ Pr α Γ Pr − a Pr − a Γ Pr α 3.31 − a α α xPr Journal of Inequalities and Applications 11 Thus, α a α xn α α xPr − xPr−1 a I1 α α xPr − xPr−1 α xn ⎛ α α xn a α xPr−1 a α α xPr ⎞ 1 ⎝ α α a α xPr−1 − α a ⎠ α xPr 3.32 Finally, α n xn dnk ⎡ a α − a α xPr k a a α xn α − a α 1 α k Pr xk α k−a xn n−a α xn ⎞⎤ ⎛ ⎝ 1 ⎠⎦ ⎣ − a − α a α xα xn Pr α α − a ⎦− a α α α a α xn − xn − xPr α xPr ⎣ ⎤ n−1 ⎡ α α α xPr a α α xn α 3.33 Clearly, D has null columns It remains to show that D has finite norm α α For all integers, n ≥ a 1, xn is positive and 1/2 ≤ qn ≤ From 3.25 , Pn −1 Pn dPn, Pn−1 dPn, k k Pn−1 ⎡ a α qn α dPn, Pn dPn, k k α α α xPn−1 α ⎣ α 3.34 ⎤ ⎦ xPn a α α Since |x Pn | ≥ 2|x Pn−1 |, then, xPn−1 /xPn ≤ 1/2, and the above sum is bounded by 4α From 3.29 , n |dnk | k α α 2a α xn α α xPr − xPr−1 a 2a α 4a α α xn α xPr α α − a α 3.35 α From choice of n, |xn | < 2|xPr | Again, using the fact that |xPr | ≥ 2|xPr−1 |, we have n k |dnk | < α α xPr a α xPr α − xPr /2 4a α 2a α 14 a α 3.36 12 Journal of Inequalities and Applications Since there are only a finite number of rows of D with n < a regular α α Proof of Case II If a is a positive integer, μa 0, and Hμ fails to have a two-sided inverse α fnk with fa,a However, if we define a new matrix F 1, D has finite norm and is and which agrees with Hμ α elsewhere, then F does possess a unique two-sided inverse Morever, F k > a, −1 fnk α −1 hnk , α hnk −1 where the α xn Hμ and, for are computed using 3.11 and 3.12 From 1.5 , for ≤ n < a Consequently, P0 0, P1 a and P2 a Now, let enk To prove that E is regular, we are concerned with the behavior of the enk E : AF −1 α −1 for all n sufficiently large We will restrict our attention to n > a Since fnk hnk −1 for all k > a, it is clear that enk dnk for k > a If we can show that enk for all ≤ k ≤ a and n > a 1, then it will follow that E is regular, since D is For n > a 1, n j a −1 fna faa n −1 − fnj fja j a α −1 −1 fnj fja 3.37 n−1 − 0, j a −1 α hnj α hja − α hnn α −1 α hna α −1 Since fa,a and hnn hna − a α / n−a , fna xn /Γ a α By induction α α α −1 it is showed that fn,a−r kr a xn , where kr a is a function of a For n > a 1, Pn−1 > Pa ≥ P1 a ≥ r Pn ePn ,r j r −1 aPn ,j fjr −1 bn,n−1 fPn−1 ,r −1 bn,n fPn ,r ⎞ ⎛ α xPn−1 ⎝ α α α α ka−r a xPn−1 − α xPn ka−r a ⎠ α qn xPn For n > a 3.38 1, n / Pi for any integer i, ≤ r ≤ a, and s the integer such that Ps < n < Ps , n en,r j r −1 an,j fjr −1 an,Ps−1 fPs−1 ,r −1 an,n fn,r 3.39 α xn α −1 an,Ps fPs ,r α α xPs − xPs−1 α α α α ka−r a xPs−1 − ka−r a xPs α ka−r a xn α Proof of Case III If a is complex, then none of the μn vanish, and we may use the matrix D of Case I It will be sufficient to show that D has finite norm From 3.25 , Pn k |dPn ,k | − a α α qn a α Pn α qn α α xPn−1 α Pn − a xPn Pn −1 |dPn ,k | k Pn−1 |dPn ,Pn−1 | 3.40 Journal of Inequalities and Applications α 13 α α Again, |xPn | ≥ 2|xPn−1 | It can be shown that 1/2 ≤ |qn | ≤ 3/2 Since a α a α α qn |dPn ,Pn−1 | Pn−1 α , 3.41 the first two and last terms of 3.40 , are clearly bounded in n For Pn−1 < k < Pn , using 3.16 , a |dPn ,k | α a Γ Pn−1 α Γ k−a 1−a Γ k Γ Pn−1 a 1 α qn α α a k − a − · · · Pn−1 − a α Γk a ≤ α a |Pn−1 a α Γ Pn−1 α − a| |Pn−1 α a Γ Pn−1 α α 3.42 |Pn−1 − a| · · · |Pn−1 Γk α Γ Pn−1 α α qn Pn −1 1 α qn α qn · where w write α Γ |Pn−1 1 − a| k − Pn−1 − Γk w , Γk α α − a| − a| − Pn−1 − w < for all n sufficiently large From Lemma 2.1, we can |dPn ,k | ≤ a α a α qn Γ |Pn−1 k Pn−1 < Γ Pn−1 α a α a 1 − a| Γ Pn−1 α Γx w w−α Γ x α α α qn Γ |Pn−1 α − a| Pn Pn−1 3.43 Γ Pn−1 w , α − w Γ Pn−1 α and the sum is uniformly bounded in n, since −w is bounded away from zero If n / Pi , for any i, then from 3.29 , n k |dnk | Pr −1 α − a α xn α α a α xn xPr − xPr−1 k Pr−1 α α a a α α a α α α α α xPr − xPr−1 xk α a α α α α xPr−1 xn α k−a α xPr−1 xn 3.44 α xPr − xPr−1 xPr − xPr−1 xPr Pr − a n−1 α a α α xk k−a k Pr a α xn −a α n n−a α 14 Journal of Inequalities and Applications Terms 1, 3, 4, and of 3.44 are clearly bounded in n Recalling that qr first summation may be written in the form a α a α α α qr xPr α xn α α − xPr−1 /xPr , the α Pr −1 k Pr−1 α xPr−1 α xk k−a 3.45 The summation is identical with the one in 3.40 , and the above expression is uniformly α α bounded, since |xn | < 2|xPr | Using an argument similar to the one used in establishing 3.40 , the second summation of 3.44 can be shown to be uniformly bounded Acknowledgment The first author acknowledges support from the Scientific and Technical Research Council of Turkey in the preparation of this paper, the authors wish to thank the referee for his careful reading of the manuscript and for his helpful suggestions References K Endl, “Abstracts of short communications and scientific program,” International Congress of Mathematicians, vol 73, p 46, 1958 K Endl, “Untersuchungen uber Momentenprobleme bei Verfahren vom Hausdorschen Typus, ă Mathematische Annalen, vol 139, pp 403432, 1960 A Jakimovski, “The product of summability methods; new classes of transformations and their properties,” Tech Note, Contract no AF61 052 -187, 1959 G H Hardy, Divergent Series, Clarendon Press, Oxford, UK, 1949 E Hille and R S Phillips, Functional Analysis and Semi-Groups, vol 31 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 1957 E Hille and J D Tamarkin, “On moment functions,” Proceedings of the National Academy of Sciences of the United States of America, vol 19, pp 902–908, 1933 E Hille and J D Tamarkin, “Questions of relative inclusion in the domain of Hausdorff means,” Proceedings of the National Academy of Sciences of the United States of America, vol 19, pp 573–577, 1933 L L Silvermann and J D Tamarkin, “On the generalization of Abel’s theorem for certain definitions of summability,” Mathematische Zeitschrift, vol 29, pp 161–170, 1928 B E Rhoades, “Size of convergence domains for known Hausdorff prime matrices,” Journal of Mathematical Analysis and Applications, vol 19, pp 457–468, 1967 10 K Zeller, “Merkwurdigkeiten bei Matrixverfahren; Einfolgenverfahren,” Archiv fur Mathematische, ă ă vol 4, pp 15, 1953 11 M R Parameswaran, “Remark on the structure of the summability field of a Hausdorff matrix,” Proceedings of the National Institute of Sciences of India Part A, vol 27, pp 175–177, 1961 12 H T Davis, The Summation of Series, The Principia Press of Trinity University, San Antonio, Texas, USA, 1962  ... multiple of 1.5 , so it is obvious that each Journal of Inequalities and Applications α −1 column of Hμ belongs to the convergence domain of A However, it will be necessary to calculate the terms of. .. convenience of regularity, we will use the sequence α μn since the constant −1/ a − a n−a α n α , 1.6 α α does not affect the size of the convergence domain of Hμ Journal of Inequalities and Applications. .. 3.40 Journal of Inequalities and Applications α 13 α α Again, |xPn | ≥ 2|xPn−1 | It can be shown that 1/2 ≤ |qn | ≤ 3/2 Since a α a α α qn |dPn ,Pn−1 | Pn−1 α , 3.41 the first two and last terms of