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Ultrametric Summability Theory

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An Introduction to Ultrametric Summability Theory

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Abstract

In this chapter, our survey of the literature on “ultrametric summability theory” starts with a paper of Andree and Petersen of 1956 (it was the earliest known paper on the topic) to the present. Most of the material discussed in the survey have not appeared in book form earlier. Silverman-Toeplitz theorem is proved using the “sliding-hump method”. Schur’s theorem and Steinhaus theorem also find a mention. Core of a sequence and Knopp’s core theorem are discussed. It is proved that certain Steinhaus type theorems fail to hold. It is also shown that the Mazur-Orlicz theorem and Brudno’s theorem fail to hold. Certain special summability methods—the N\(\ddot{\mathrm{o}}\)rlund method, Weighted mean method, \(Y\)-method, \(M\)-method, Euler method and Taylor method are introduced and their properties are extensively discussed. Some product theorems and Tauberian theorems are proved. Double sequences and double series are introduced and Silverman-Toeplitz theorem for four-dimensional infinite matrices is proved. Some applications of this theorem are discussed towards the end of the chapter.

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  1. Department of Mathematics, Ramakrishna Mission Vivekananda College, Chennai, Tamil Nadu, India

    P. N. Natarajan

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Natarajan, P.N. (2014). Ultrametric Summability Theory. In: An Introduction to Ultrametric Summability Theory. SpringerBriefs in Mathematics. Springer, New Delhi. https://doi.org/10.1007/978-81-322-1647-6_4

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