{ Cnk } n The matrix A is said to be regular if every convergent sequence is summable-A to its natural limit; for this the requirements on the aik are the celebrated Silverman-Toeplitz conditions [1, p. 64]. The idea of the present paper derives from the appeal to replace the application of the natural limit in (1) in both instances by summability-A itself, thereby yielding for regular A a not-less-general transform. More generally we consider the succession of functionals defined by repetitions of this double iteration. DEFINITION 1. With respect to a matrix A = (aik), the A I-Operator of order n, Wn, for n=O, 1, 2, * * *, is the functional, operating on sequences, defined by the following recursion: Wo{ Xk } = limk-. Xk, and Wn+l{xk}= Wn Wn{EJk=o aikXk} i} i. Thus W1{xk} is the usual A-sum; and by way of example:
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