By P.N. Natarajan

ISBN-10: 8132216466

ISBN-13: 9788132216469

ISBN-10: 8132216474

ISBN-13: 9788132216476

Ultrametric research has emerged as an incredible department of arithmetic in recent times. This publication offers, for the 1st time, a short survey of the study so far in ultrametric summability thought, that is a fusion of a classical department of arithmetic (summability thought) with a contemporary department of research (ultrametric analysis). numerous mathematicians have contributed to summability conception in addition to sensible research. The e-book will attract either younger researchers and more matured mathematicians who're seeking to discover new components in analysis.

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**Example text**

K = R or C or a complete, non-trivially valued, ultrametric field, r is the space of all sequences x = {xk } ∈ ∞ such that |xk+r − xk | → 0, k → ∞, r ≥ 1 being a fixed integer. It is easily proved that r is a closed subspace of ∞ with respect to the norm defined for elements in ∞ . When K = R or C, the following result, improving 36 4 Ultrametric Summability Theory Steinhaus theorem, was proved in [10] (it is worthwhile to note that a constructive proof was given). 4 (c, c; P) ∩ ( r r −1 − i , c) i=1 = φ.

N Then f is continuous everywhere but not differentiable anywhere (see [3]). In classical analysis, functions which have antiderivatives do not have jump discontinuities and they are pointwise limits of continuous functions. However, both these conditions are not sufficient for the functions to have an antiderivative. Unlike the classical case in which sufficient conditions are not known, the situation in the ultrametric case is simpler: If U is a subset of K without isolated points, then f : U → K has an antiderivative if and only if f is the pointwise limit of continuous functions ([4], p.

However, both these conditions are not sufficient for the functions to have an antiderivative. Unlike the classical case in which sufficient conditions are not known, the situation in the ultrametric case is simpler: If U is a subset of K without isolated points, then f : U → K has an antiderivative if and only if f is the pointwise limit of continuous functions ([4], p. 283). References 1. : Introduction to p-adic Numbers and Valuation Theory. Academic Press, New York (1964) 2. : Ultrametric Calculus.

### An Introduction to Ultrametric Summability Theory by P.N. Natarajan

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