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During a recent conversation, R. P. Agnew suggested a determination of the validity of the proposition that row-infinite Toeplitz transformations are more powerful than row-finite transformations. Before this proposition is examined, it is necessary to assign a precise meaning to it. Corresponding to every sequence s a regular rowfinite Toeplitz transformation A can be constructed such that the transform As converges to a finite limit. In order to be of interest, the proposition must therefore be interpreted in terms of convergence fields of individual transformations. Also, because every row-infinite regular Toeplitz matrix is the sum of two Toeplitz matrices A and B, where A is regular and row-finite and B has the norm zero, the convergence field in the space of bounded sequences of every regular Toeplitz transformation coincides with that of a row-finite regular transformation; in other words, the problem is trivial except in its reference to unbounded sequences.

Convergence fields of row-finite and row-infinite Toeplitz transformations