New PDF release: An Introduction to Ultrametric Summability Theory

By P.N. Natarajan

ISBN-10: 8132216466

ISBN-13: 9788132216469

ISBN-10: 8132216474

ISBN-13: 9788132216476

Ultrametric research has emerged as a big department of arithmetic lately. This publication offers, for the 1st time, a short survey of the examine thus far in ultrametric summability thought, that's a fusion of a classical department of arithmetic (summability thought) with a contemporary department of study (ultrametric analysis). numerous mathematicians have contributed to summability conception in addition to sensible research. The publication will attract either younger researchers and more matured mathematicians who're seeking to discover new parts in analysis.

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We can choose (1) (2) (1) (2) two subsequences of x (say) {xk }, {xk } such that if yk = xk − xk , yk = 0, k = p while y p = 0. Such choice is possible since x diverges and therefore has two unequal entries after any stage for k. The sequence {yk } is summable A. So it follows that lim anp exists, p = 0, 1, 2, . . Next we show that lim anp = 0, n→∞ p→∞ n = 0, 1, 2, . . For, otherwise, there exists ε > 0 and a non-negative integer m such that |am, p(i) | > ε , i = 1, 2, . . , 48 4 Ultrametric Summability Theory where { p(i)} is an increasing sequence of positive integers.

If lim (Ax)n = , we say that n→∞ the sequence x = {xk } is A-summable or summable A to . 2 Let X, Y be sequence spaces with elements whose entries are in K . The infinite matrix A = (ank ), ank ∈ K , n, k = 0, 1, 2, . . is said to transform X to Y if whenever the sequence x = {xk } ∈ X , (Ax)n is defined, n = 0, 1, 2, . . and the sequence {(Ax)n } ∈ Y . In this case, we write A ∈ (X, Y ). 3 If A ∈ (c, c) (where c is the ultrametric Banach space consisting of all convergent sequences in K with respect to the norm defined by ||x|| = sup |xk |, k≥0 x = {xk } ∈ c), A is said to be convergence preserving or conservative.

It now follows that {xα(k) } is not summable A, a contradiction. Thus, in both cases it turns out that x is bounded, if A were not to be a Schur matrix. Next we observe that since A is not a Schur matrix (see [9]), there exist ε > 0 and an increasing sequence {n(i)} of positive integers such that sup |an(i)+1, p − an(i), p | > ε, i = 1, 2, . . p≥0 Hence there exists p(n(i)) such that |an(i)+1, p(n(i)) − an(i), p(n(i)) | > ε, i = 1, 2, . . 26) Suppose { p(n(i))} is bounded, then there are only a finite number of distinct entries in that sequence.

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


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