与Walsh系统相关的算子序列的极限
Limits of sequences of operators associated with Walsh System
AI总结：
本文研究Walsh系统相关算子序列及张量积算子序列的收敛性，给出权重条件以确保依测度收敛，并指出几乎处处收敛的局限性。
摘要：
本文旨在确定权重$\mathbf{q}=\{q_k\}$的必要且充分条件，以保证与Walsh系统相关的算子序列$\left\{ T_{n}^{\left( \mathbf{q}\right) }f\right\} $对所有可积函数$f$几乎处处收敛。文章还考察了涉及二元函数的张量积算子序列$\left\{ T_{n}^{( \mathbf{q})}\otimes T_{n}^{( \mathbf{p})}\right\}$的收敛性。我们指出，Gát和Karagulyan（2016）的最新研究表明，该张量积算子序列不能对每个可积函数几乎处处收敛。本文提供了权重的必要且充分条件，以确保上述算子序列在$L_{1}$上依测度收敛。
英文摘要：
The aim of the current paper is to determine the necessary and sufficient conditions for the weights $\mathbf{q}=\{q_k\}$, ensuring that the sequence of operators $\left\{ T_{n}^{\left( \mathbf{q}\right) }f\right\} $ associated with Walsh system, is convergent almost everywhere for all integrable function $f$. The article also examines the convergence of a sequence of tensor product operators denoted as $\left\{ T_{n}^{( \mathbf{q})}\otimes T_{n}^{( \mathbf{p})}\right\}$ involving functions of two variables. We point out that recent research by Gát and Karagulyan (2016) demonstrated that this sequence of tensor product operators cannot converge almost everywhere for every integrable function. In this paper, the necessary and sufficient conditions for the weight are provided which ensure that the sequence of the mentioned operators converges in measure on $L_{1}$.
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