Matrix transformations and Walsh's equiconvergence theorem (Q2368382)

Summary: In 1977, \textit{R. T. Jacob} [Pac. J. Math. 70, 179--187 (1977; Zbl 0387.46007)] defined \(G_{\alpha}\), for any \(0 \leq \alpha < \infty\), as the set of all complex sequences \(x\) such that lim sup \(|x_k|^{1/k} \leq \alpha\). In this paper, we apply the \(G_u - G_v\) matrix transformation on the sequences of operators given in the famous Walsh equiconvergence theorem [\textit{J. L. Walsh}, Interpolation and approximation by rational functions in the complex domain (American Mathematical Society. Colloquium Publications 20, Providence, R.I.) (1965; Zbl 0146.29902)], where we have that the difference of two sequences of operators converges to zero in a disk. We show that the \(G_u - G_v\) matrix transformation of the difference converges to zero in an arbitrarily large disk. Also, we give examples of such matrices.