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Some Tauberian theorems for four-dimensional Euler and Borel summability
Advances in Difference Equations volume 2015, Article number: 50 (2015)
Abstract
The four-dimensional summability methods of Euler and Borel are studied as mappings from absolutely convergent double sequences into themselves. Also the following Tauberian results are proved: if \(x=(x_{m,n})\) is a double sequence that is mapped into \(\ell_{2}\) by the four-dimensional Borel method and the double sequence x satisfies \(\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}|\Delta_{10} x_{m,n}|\sqrt {mn}<\infty\) and \(\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}|\Delta_{01} x_{m,n}|\sqrt {mn}<\infty\), then x itself is in \(\ell_{2}\).
1 Introduction
The best-known notion of convergence for double sequences is convergence in the sense of Pringsheim. Recall that a double sequence \(x=\{x_{k,l}\}\) of complex (or real) numbers is called convergent to a scalar L in the sense of Pringsheim (denoted by \(P\mbox{-}\!\lim x=L\)) if for every \(\epsilon> 0\) there exists an \(N \in\mathbb{N}\) such that \(\vert x_{k,l} - L\vert < \epsilon\) whenever \(k,l > N\). Such an x is described more briefly as ’P-convergent’. It is easy to verify that \(x=\{x_{k,l}\}\) converges in the sense of Pringsheim if and only if for every \(\epsilon> 0\) there exists an integer \(N=N(\epsilon)\) such that \(\vert x_{i,j}-x_{k,l} \vert < \epsilon\) whenever min\(\{i,j,k,l\}\geq N\). A double sequence \(x=\{x_{k,l}\}\) is bounded if there exists a positive number M such that \(|x_{m,n}|\leq M\) for all m and n, that is, if \(\sup_{m,n}|x_{m,n}|<\infty\).
A double sequence \(x=\{x_{k,l}\}\) is said to convergence regularly if it converges in the sense of Pringsheim and, in addition, the following finite limits exist:
Let \(A=(a_{m,n,k,l})\) denote a four-dimensional summability method that maps the complex double sequence x into the double sequence Ax where the mnth term of Ax is as follows:
In [1] Robison presented the following notion of regularity for four-dimensional matrix transformation and a Silverman-Toeplitz type characterization of such a notion.
Definition 1.1
The four-dimensional matrix A is said to be RH-regular if it maps every bounded P-convergent sequence into a P-convergent sequence with the same P-limit.
The assumption of boundedness was added because a double sequence which is P-convergent is not necessarily bounded. Along these same lines, Robison and Hamilton presented a Silverman-Toeplitz type multidimensional characterization of regularity in [2] and [1].
Theorem 1.1
The four-dimensional matrix A is RH-regular if and only if
- \(RH_{1}\)::
-
\(P\mbox{-}\!\lim_{m,n}a_{m,n,k,l} = 0\) for each k and l;
- \(RH_{2}\)::
-
\(P\mbox{-}\!\lim_{m,n}\sum_{k,l=0,0}^{\infty,\infty }a_{m,n,k,l} = 1\);
- \(RH_{3}\)::
-
\(P\mbox{-}\!\lim_{m,n}\sum_{k=0}^{\infty} \vert a_{m,n,k,l}\vert = 0\) for each l;
- \(RH_{4}\)::
-
\(P\mbox{-}\!\lim_{m,n}\sum_{l=0}^{\infty} \vert a_{m,n,k,l}\vert = 0\) for each k;
- \(RH_{5}\)::
-
\(\sum_{k,l=0,0}^{\infty,\infty} \vert a_{m,n,k,l}\vert \) is P-convergent;
- \(RH_{6}\)::
-
there exist finite positive integers Δ and Γ such that \(\sum_{k,l>\Gamma} \vert a_{m,n,k,l}\vert <\Delta\).
The set of all absolutely convergent double sequences will be denoted \(\ell_{2}\), that is,
For single sequences, in [3] Fridy and Roberts proved the following Tauberian theorem.
Theorem 1.2
If B is a Borel matrix and \(x=(x_{k})\) is a sequence such that Bx in \(\ell=\{x=(x_{k}): \sum_{k=1}^{\infty }|x_{k}|<\infty\}\) and
then x is in â„“.
Our aim is to extend the results in [3] from single absolutely convergent sequences to double absolutely convergent sequences. In [4], Patterson proved that the matrix \(A=(a_{m,n,k,l})\) determines an \(\ell_{2}\mbox{-}\ell_{2}\) method if and only if
2 Euler-Knopp and Borel \(\ell_{2}\mbox{-}\ell_{2}\) methods
The four-dimensional Euler-Knopp method, for any complex numbers \(r_{1}\) and \(r_{2}\), is defined by
An application of the Maclaurin series expansion of \((1-z_{1})^{k+1}(1-z_{2})^{l+1}\) shows that each column sum of \(E_{r_{1},r_{2}}\) converges absolutely to \(\frac{1}{r_{1}r_{2}}\) provided that \(0< r_{1}\leq1\) and \(0< r_{2}\leq1\). If \(0< r_{1}< 1\) and \(0< r_{2}< 1\), then \(P\mbox{-}\!\lim_{m,n} E_{r_{1},r_{2}}[m,n,m,n]=0\), so \(E_{r_{1},r_{2}}^{-1}\) is not an \(\ell_{2}\mbox{-}\ell_{2}\) matrix. We summarize this as follows.
Theorem 2.1
The four-dimensional Euler-Knopp method \(E_{r_{1},r_{2}}\) is a sum-preserving \(\ell_{2}\mbox{-}\ell_{2}\) matrix for which \(\ell_{2_{E_{r_{1},r_{2}}}}\neq\ell_{2}\) if and only if \(0< r_{1}< 1\) and \(0< r_{2}< 1\), where \(\ell_{2_{E_{r_{1},r_{2}}}}\) is the summability field of \(E_{r_{1},r_{2}}\).
The four-dimensional Borel method B is given by the matrix
By a direct application of Theorem 3.1 in [4], one can show that B is an \(\ell_{2}\mbox{-}\ell_{2}\) matrix.
Theorem 2.2
If \(r_{1}>0\) and \(r_{2}>0\) and \(x=(x_{k,l})\) is a double sequence such that \(E_{r_{1},r_{2}}x\) is in \(\ell_{2}\), then Bx is in \(\ell_{2}\).
Proof
We use the familiar technique of showing that \(BE_{r_{1},r_{2}}\) is an \(\ell_{2}\mbox{-}\ell_{2}\) matrix. Since \(Bx=BE_{r_{1},r_{2}}^{-1}E_{r_{1},r_{2}}x\), this will ensure that Bx is in \(\ell_{2}\) whenever \(E_{r_{1},r_{2}}x\) in \(\ell_{2}\). Since \(E_{r_{1},r_{2}}^{-1}=E_{\frac{1}{r_{1}},\frac{1}{r_{2}}}\) we replace \(s_{1}=\frac{1}{r_{1}}\) and \(s_{2}=\frac{1}{r_{2}}\) and show that \(BE_{s_{1},s_{2}}\) is an \(\ell_{2}\mbox{-}\ell_{2}\) matrix for all positive \(s_{1}\) and \(s_{2}\). The \(mnkl\)th term of \(BE_{s_{1},s_{2}}\) is given by
Summing the \((k,l)\)th column of \(BE_{s_{1},s_{2}}\), we get
Hence,
so \(BE_{s_{1},s_{2}}\) is an \(\ell_{2}\mbox{-}\ell_{2}\) matrix. □
Theorem 2.2 and the \(\ell_{2}\mbox{-}\ell_{2}\) property of \(E_{r_{1},r_{2}}\) lead to the following result.
Theorem 2.3
The four-dimensional Borel matrix determines an \(\ell_{2}\mbox{-}\ell_{2}\) method.
In addition to the inclusion relation given in Theorem 2.2, we can also show that the \(\ell_{2}\mbox{-}\ell_{2}\) method B is strictly stronger than all \(E_{r_{1},r_{2}}\) methods by the following example.
Example 2.1
Suppose \(r_{1}>0\) and \(r_{2}>0\) and \(x_{k,l}=(-s_{1})^{k}(-s_{2})^{l}\) where \(s_{1}\geq-1+\frac{2}{r_{1}}\) and \(s_{2}\geq-1+\frac {2}{r_{2}}\); then Bx is in \(\ell_{2}\) but \(E_{r_{1},r_{2}}\) is not in \(\ell_{2}\). Let us consider the following methods:
and
By solving \(-1< 1-r_{1}-r_{1}s_{1}<1\) and \(-1< 1-r_{2}-r_{2}s_{2}<1\), we see that \(E_{r_{1},r_{2}}x\) is in \(\ell_{2}\) if and only if \(-1< s_{1}<-1+\frac{2}{r_{1}}\) and \(-1< s_{2}<-1+\frac{2}{r_{2}}\).
3 Tauberian theorems
To prove Theorem 3.1 we need the following lemma.
Lemma 3.1
If
and r and s are positive integers, then
-
(i)
$$\sum_{m=r+1}^{\infty}\sum _{n=s+1}^{\infty}\sum_{k=0}^{r} \sum_{l=0}^{s}b_{m,n,k,l}=O(\sqrt{rs}) $$
and
-
(ii)
$$\sum_{m=0}^{r}\sum _{n=0}^{s}\sum_{k=r+1}^{\infty} \sum_{l=s+1}^{\infty}b_{m,n,k,l}=O(\sqrt{rs}). $$
Proof
Let \(p=[\sqrt{r}]\) and \(q=[\sqrt{s}]\), and let us write the sum in (i) as
If \(s_{1}< m\) and \(s_{2}< n\), then
In \(\phi_{r,s}\), let \(s_{1}=r-p\), \(s_{2}=s-q\), and
thus
In \(\varphi_{r,s}\), observe that
thus
Hence, (i) is proved. Next write the sum in (ii) as
Assume that \(\lambda_{r,s}=0\) if \(p=1\), \(q=1\). Then
If \(s_{1}\geq m\) and \(s_{2}\geq n\), then
Let \(s_{1}=r+p\) and \(s_{2}=s+q\), we have
Thus the lemma is proved. □
We are now ready to prove the following result.
Theorem 3.1
If x is a double sequence such that Bx is in \(\ell_{2}\),
and
then x in \(\ell_{2}\) where \(\Delta_{10}x_{r,s}=x_{r,s}-x_{r+1,s}\) and \(\Delta_{01}x_{r,s}=x_{r,s}-x_{r,s+1}\).
Proof
It is suffices to show that \(Bx-x\) is in \(\ell_{2}\); that is,
Since
for each m, n, the above sum can be written as
and we need only show the following:
Let \(S=S_{1}+S_{2}\), where
and
Since
Also,
By Lemma 3.1, \(\zeta_{r,s}=O(\sqrt{rs})\) and \(\varsigma_{r,s}=O(\sqrt{rs})\), we have
which proves the theorem. □
Combining Theorem 3.1 with Theorem 2.2, we are lead to the following \(\ell_{2}\mbox{-}\ell_{2}\) Tauberian theorem for the four-dimensional Euler-Knopp means.
Theorem 3.2
If \(r_{1}>0\), \(r_{2}>0\), and x is a double sequence satisfying (3.1) such that \(E_{r_{1},r_{2}}\) is in \(\ell_{2}\), then x is in \(\ell_{2}\).
Example 3.1
The following double sequence is not mapped into \(\ell_{2}\) by B or by \(E_{r_{1},r_{2}}\), with \(r_{1}>0\), \(r_{2}>0\). Define \(x=\{x_{k,l}\}\) by
Then x satisfies (3.1) and (3.2), but x is not in \(\ell_{2}\) because if \(k\geq1\) and \(l\geq1\),
Hence, by Theorem 3.1, Bx is not in \(\ell_{2}\).
References
Robison, GM: Divergent double sequences and series. Trans. Am. Math. Soc. 28, 50-73 (1926)
Hamilton, HJ: Transformations of multiple sequences. Duke Math. J. 2, 29-60 (1936)
Fridy, JA, Roberts, KL: Some Tauberian theorems for Euler and Borel summability. Int. J. Math. Math. Sci. 3(4), 731-738 (1980)
Patterson, RF: Four dimensional matrix characterization of absolute summability. Soochow J. Math. 30(1), 21-26 (2004)
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Nuray, F., Patterson, R.F. Some Tauberian theorems for four-dimensional Euler and Borel summability. Adv Differ Equ 2015, 50 (2015). https://doi.org/10.1186/s13662-015-0381-2
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DOI: https://doi.org/10.1186/s13662-015-0381-2