Analogues of some Tauberian theorems for stretchings (Q1599735)

Let \(x= (x_{nk})\) be a double sequence. A double sequence \(y= (y_{nk})\) is called a subsequence of \(x\) if there exist two strictly increasing index sequences \((n_i)\) and \((k_i)\) such that \[ y= \left\{\begin{matrix} z_1& z_2 & z_5 & z_{10} & \cdot\\ z_4 & z_3 & z_6 & \cdot & \cdot\\ z_9 & z_8 & z_7 & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot & \cdot\end{matrix}\right. \] where \(z_i:= x_{n_i k_i}\). For a double sequence \(\varepsilon= (\varepsilon_{ij})\), the double sequence \(y\) is said to contain an \(\varepsilon\)-Pringsheim-copy of \(x\) if \(y\) contains a subsequence \((y_{n_i k_j})\) with \(|y_{n_i k_j}- x_{ij}|< \varepsilon_{ij}\) \((i,j\in\mathbb{N})\). A double sequence \(y\) is called a stretching of \(x\) if there exist two strictly increasing index sequences \((R_i)^\infty_{i=0}\) and \((S_j)^\infty_{j=0}\) such that \(R_0= S_0= 1\) and \[ y_{nk}:= \begin{cases} x_{ni},\quad &\text{if }R_{i-1}\leq k< R_i,\\ x_{jk},\quad &\text{if }S_{j-1}\leq n< S_j\end{cases}\qquad (i,j\in\mathbb{N}). \] A four-dimensional matrix \(A= (a_{mnkl})\) is called RH-regluar if it maps every bounded \(P\)-convergent (\(P\) for Pringsheim) double sequence into a \(P\)-convergent double sequence with the same \(P\)-limit. The main result of the paper is the following extended copy theorem [cf. \textit{D. F. Dawson}, Pac. J. Math. 77, 75-81 (1978; Zbl 0393.40007)]. Theorem. Suppose \(A= (a_{mnkl})\) and \(T= (t_{mnkl})\) are RH-regular matrices. If \(x\) is bounded, i.e. \(\sup_{n,k}|x_{nk}|< \infty\), and \(\varepsilon\) is any bounded positive term double sequence with \(P\)-\(\lim\varepsilon_{ij}= 0\), then there exists a stretching \(y\) of \(x\) such that \(T(Ay)\) exists and contains an \(\varepsilon\)-Pringsheim-copy of \(x\). Corollary. Let \(T\) be a RH-regular matrix and let \(A\) be a RH-regular matrix such that \(Ay\) is (absolutely) \(T\)-summable for every stretching \(y\) of \(x\). Then \(x\) is \(P\)-convergent.