Read e-book online An Introduction to Ultrametric Summability Theory PDF

By P.N. Natarajan

ISBN-10: 8132225589

ISBN-13: 9788132225584

ISBN-10: 8132225597

ISBN-13: 9788132225591

This is the second one, thoroughly revised and multiplied version of the author’s first ebook, protecting various new themes and up to date advancements in ultrametric summability conception. Ultrametric research has emerged as a massive department of arithmetic in recent times. This booklet provides a quick survey of the learn so far in ultrametric summability idea, that is a fusion of a classical department of arithmetic (summability concept) with a latest department of research (ultrametric analysis). numerous mathematicians have contributed to summability conception in addition to sensible research. The e-book will entice either younger researchers and more matured mathematicians who're trying to discover new parts in research. The e-book is usually beneficial as a textual content should you desire to specialise in ultrametric summability theory.

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Read e-book online An Introduction to Ultrametric Summability Theory PDF

This is often the second one, thoroughly revised and extended version of the author’s first e-book, masking a number of new themes and up to date advancements in ultrametric summability thought. Ultrametric research has emerged as a big department of arithmetic lately. This publication offers a quick survey of the study thus far in ultrametric summability conception, that is a fusion of a classical department of arithmetic (summability thought) with a latest department of research (ultrametric analysis).

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0, k = k(i) ∞ |xk(i) |α ≤ i=1 ∞ i=1 1 < ∞. 2 Steinhaus-Type Theorems 47 where n(m) 1 |an,k(m) |α |xk(m) |α , = n=n(m−1)+1 n(m) 2 m−1 = |an,k(i) |α |xk(i) |α , n=n(m−1)+1 i=1 ∞ n(m) 3 |an,k(i) |α |xk(i) |α . = n=n(m−1)+1 i=m+1 Now, n(m) 1 |an,k(m) |α ρ(λ(m)+1)α = n=n(m−1)+1 n(m) ≥ ρα |an,k(m) |α m −2 n=n(m−1)+1 m−1 >2+ i −2 μk(i) ; i=1 n(m) 2 m−1 = |an,k(i) |α ρ(λ(i)+1)α n=n(m−1)+1 i=1 m−1 ≤ n(m) i −2 i=1 m−1 ≤ |an,k(i) |α n=n(m−1)+1 i −2 μk(i) ; i=1 n(m) 3 ∞ = |an,k(i) |α ρ(λ(i)+1)α n=n(m−1)+1 i=m+1 n(m) ∞ ≤ n=n(m−1)+1 i=k(m+1) < 1.

Whatever be K , it is well known that an infinite matrix which sums all sequences of 0’s and 1’s sums ∞ all bounded sequences (see [14, 15]). It is clear that any Cauchy sequence is in so that each r r =1 r is a sequence space containing the space C of Cauchy sequences. It ∞ may be noted that C ∞ r when K = R or C while C = r =1 r when K is a r =1 complete, non-trivially valued, ultrametric field. Although r do not form a tower between C and ∞ , they can be deemed to reflect the measure of non-Cauchy nature of sequences contained in them.

It is clear that A is regular. Consequently, (c, c; P) ∩ ( r , c) = φ, proving our claim. We note that (c, c; P) ∩ ( r , c) = φ when K = R or C. Since ( ∞ , c) ⊆ ( r , c), it follows that (c, c; P) ∩ ( ∞ , c) = φ, which is Steinhaus theorem. We call such results Steinhaus-type theorems. For more Steinhaus-type theorems, see [11–13]. Let us now see in detail the role played by the sequence spaces r . Whatever be K , it is well known that an infinite matrix which sums all sequences of 0’s and 1’s sums ∞ all bounded sequences (see [14, 15]).

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


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