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Core-Consistency and Total Inclusion for Methods of Summability

  • G. G. Lorentz (a1) and A. Robinson (a2)

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We shall consider methods of summation A, B, … defined by matrices of real elements (amn), (bmn), (m, n = 1, 2, …) which are regular, that is, have the three well-known properties of Toeplitz (4, p. 43). A method A is said to be core-consistent with the method B for bounded sequences if the A-core (3, p. 137; and 4, p. 55) of each real bounded sequence is contained in its B-core. B is totally included in A, B ≪A, if each real sequence which is B-summable to a definite limit (this limit may be finite or infinite of a definite sign) is also A-summable to the same limit.

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References

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1. Agnew, R. P., Cores of complex sequences and of their transforms, Amer. J. Math. 61 (1939), 178–186.
2. Basu, S. K., On the total relative strength of the Hölder and Cesàro methods, Proc. London Math. Soc. (2), 50 (1949), 447–462.
3. Cooke, R. G., Infinite matrices and sequence spaces (London, 1950).
4. Hardy, G. H., Divergent series (Oxford, ).
5. Hurwitz, W. A., Some properties of methods of evaluation of divergent sequences, Proc. London Math. Soc. (2), 26 (1926), 231–248.
6. Knopp, K., Zur Theorie der Limitierungsverfahren. Math. Zeitschrift, 31 (1929-30), pp. 97–127, 276–305.
7. Krein, M. G. and Rutman, M. A., Linear operators leaving invariant a cone in a Banach space, Uspehi Mat. Nauk (N.S.), 3, no 23 (1948), 3–95; Amer. Math. Soc. Translations no. 26 (1950).
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Core-Consistency and Total Inclusion for Methods of Summability

  • G. G. Lorentz (a1) and A. Robinson (a2)

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