Size of Convergence Domains for Generalized Hausdorff Prime Matrices
Journal of Inequalities and Applications volume 2011, Article number: 131240 (2011)
We show that there exit E-J generalized Hausdorff matrices and unbounded sequences such that each matrix has convergence domain .
The convergence domain of an infinite matrix will be denoted by and is defined by , where denotes the space of convergence sequences, . The necessary and sufficient conditions of Silverman and Toeplitz for a matrix to be conservative are exists for each , exists, and . A conservative matrix is called multiplicative if each and regular if, in addition, .
where is real number, is a real or complex sequence and is forward difference operator defined by . We will consider here only nonnegative . For , one obtains an ordinary Hausdorff matrix.
in which case is called the moment generating function, or mass function, for and is called moment sequence.
For ordinary Hausdorff summability , the necessary and sufficient conditions, for regularity are that function , , , and (1.2) is satisfied with .
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, 7]), using some techniques from , showed that every Hausdorff matrix with moment function
is prime. In 1967, Rhoades  showed that the convergence domain of every known prime Hausdorff matrix is of the form for a particular unbounded sequence .
Given any unbounded sequence , Zeller  constructed a regular matrix with convergence domain . It has been shown by Parameswaran  that if is any unbounded sequence such that is bounded, divergent, and Borel summable, then no Hausdorff matrix exists with .
The main result of this paper is to show that there exist E-J generalized Hausdorff matrices whose moment sequences are
and unbounded sequences such that each matrix has convergent domain .
Define the sequences by
where it is understood that if is positive integer, then for .
If is the moment sequence defined by , , then it is clear that . Hence, it will be sufficient to prove the theorem by using , in (1.4). To have the convenience of regularity, we will use the sequence
since the constant does not affect the size of the convergence domain of .
2. Auxiliary Results
In order to prove the main theorem of this paper, we will need the following results.
Let , , , , . Then, formally, for any ,
Lemma 2.1 appears as formula 12 on page 138 of .
For integers , as in (1.5),
Using Lemma 2.1,
can be written as
so that, for , . From Lemma 2.1 and (3.11),
3. Main Result
If for fixed and the matrix is defined by (1.4) and a sequence by (1.5), then .
We will first show that .
We can write the matrix , where the diagonal entries of are
For each and ,
Define , where
From Lemma 2.1,
This argument is valid provided is not a positive integer. If is a positive integer, then for .
Then, for , and for , from Lemma 2.1, we get
Since is regular, . Thus, .
To prove the converse, we will use Zeller's technique to construct a regular matrix with and then show that .
Set and define a sequence inductively by selecting to be smallest integer such that . (Such a construction is clearly possible, since is not bounded.) Let , . Define a matrix by
Now, define the matrix as follows:
If for any integer , then there exists an integer such that . For this , define
Set otherwise. From , is regular and . There are three cases to consider, based on whether is real number and not a positive integer, is positive integer, or is complex.
Proof of Case I.
If is real and not a positive integer, the E-J generalized Hausdorff matrix generated by (1.6) has a unique two sided inverse with generating sequence
To show that , it will be sufficent to show that is a regular matrix. Each column of is essentially a scalar multiple of (1.5), so it is obvious that each column of belongs to the convergence domain of . However, it will be necessary to calculate the terms of explicitly, since we must show that and that has finite norm.
If for any integer , and denotes the integer such that , then from the definition of ,
If for , then
For for any , if we now let denote the integer such that , then for
The quantity in brackets is equal to , giving
By using (3.13)–(3.17),
By using Lemma 2.2, and noting that
For for any , the integer such that , and using (3.18)–(3.24), we have
Writing and using Lemma 2.2, the quantity in brackets, which we call , takes the form
Clearly, has null columns. It remains to show that has finite norm.
For all integers, , is positive and . From (3.25),
Since , then, , and the above sum is bounded by . From (3.29),
From choice of , . Again, using the fact that , we have
Since there are only a finite number of rows of with , has finite norm and is regular.
Proof of Case II.
If is a positive integer, , and fails to have a two-sided inverse. However, if we define a new matrix with and which agrees with elsewhere, then does possess a unique two-sided inverse. Morever, and, for , , where the are computed using (3.11) and (3.12).
From (1.5), for . Consequently, and . Now, let . To prove that is regular, we are concerned with the behavior of the for all sufficiently large. We will restrict our attention to . Since for all , it is clear that for . If we can show that for all and , then it will follow that is regular, since is.
Since and . By induction it is showed that , where is a function of .
For , for any integer , , and the integer such that ,
Proof of Case III.
If is complex, then none of the vanish, and we may use the matrix of Case I. It will be sufficient to show that has finite norm. From (3.25),
Again, . It can be shown that . Since
the first two and last terms of (3.40), are clearly bounded in .
For , using (3.16),
where . for all sufficiently large. From Lemma 2.1, we can write
and the sum is uniformly bounded in , since is bounded away from zero.
If , for any , then from (3.29),
Terms , and 6 of (3.44) are clearly bounded in . Recalling that , the first summation may be written in the form
The summation is identical with the one in (3.40), and the above expression is uniformly bounded, since . 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.
Endl K: Abstracts of short communications and scientific program. International Congress of Mathematicians 1958, 73: 46.
Endl K: Untersuchungen über Momentenprobleme bei Verfahren vom Hausdorffschen Typus. Mathematische Annalen 1960, 139: 403–432. 10.1007/BF01342846
Jakimovski A: The product of summability methods; new classes of transformations and their properties. Tech. Note, Contract no. 1959., (AF61 (052)-187):
Hardy GH: Divergent Series. Clarendon Press, Oxford, UK; 1949:xvi+396.
Hille E, Phillips RS: Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications. Volume 31. American Mathematical Society, Providence, RI, USA; 1957:xii+808.
Hille E, Tamarkin JD: On moment functions. Proceedings of the National Academy of Sciences of the United States of America 1933, 19: 902–908. 10.1073/pnas.19.10.902
Hille E, Tamarkin JD: Questions of relative inclusion in the domain of Hausdorff means. Proceedings of the National Academy of Sciences of the United States of America 1933, 19: 573–577. 10.1073/pnas.19.5.573
Silvermann LL, Tamarkin JD: On the generalization of Abel's theorem for certain definitions of summability. Mathematische Zeitschrift 1928, 29: 161–170.
Rhoades BE: Size of convergence domains for known Hausdorff prime matrices. Journal of Mathematical Analysis and Applications 1967, 19: 457–468. 10.1016/0022-247X(67)90004-2
Zeller K: Merkwürdigkeiten bei Matrixverfahren; Einfolgenverfahren. Archiv für Mathematische 1953, 4: 1–5.
Parameswaran MR: Remark on the structure of the summability field of a Hausdorff matrix. Proceedings of the National Institute of Sciences of India. Part A 1961, 27: 175–177.
Davis HT: The Summation of Series. The Principia Press of Trinity University, San Antonio, Texas, USA; 1962:ix+140.
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.
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Selmanogullari, T., Savaş, E. & Rhoades, B.E. Size of Convergence Domains for Generalized Hausdorff Prime Matrices. J Inequal Appl 2011, 131240 (2011). https://doi.org/10.1155/2011/131240