By Hemen Dutta, Billy E. Rhoades

Comprises either classical and smooth tools in summability theory

Focuses at the simple advancements bearing on an idea in complete details

Integrates theories in addition to functions, at any place possible

This ebook discusses contemporary advancements in and modern study on summability conception, together with basic summability equipment, direct theorems on summability, absolute and powerful summability, particular tools of summability, sensible analytic tools in summability, and comparable issues and functions. All contributing authors are eminent scientists, researchers and students of their respective fields, and hail from round the world.

Summability idea is usually utilized in research and utilized arithmetic. It performs a huge half within the engineering sciences, and numerous points of the idea have lengthy seeing that been studied by means of researchers all around the world.

Audience

The e-book can be utilized as a textbook for graduate and senior undergraduate scholars, and as a worthy reference advisor for researchers and practitioners within the fields of summability conception and useful analysis.

Topics

Sequences, sequence, Summability

Functional Analysis

Approximations and Expansions

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**Additional resources for Current Topics in Summability Theory and Applications**

**Sample text**

E. Pn ≤ M for some M > 0, n = 0, 1, 2, . . pn Q n ∞ ∞ xn be (N , pn ) summable to . Then Let n=0 bn converges to if and only if n=0 ∞ sup |Q n | n ∞ where bn = qn k=n k=n Pk Q k+1 qk qk+2 − pk Q k pk+1 Q k+2 xk , n = 0, 1, 2, . .. N. Natarajan Proof Let n sn = xk , k=0 tn = p 0 s0 + p 1 s1 + · · · + p n sn , n = 0, 1, 2, . . Pn Then s0 = t 0 , 1 sn = (Pn tn − Pn−1 tn−1 ), n = 1, 2, . . e. lim tn = . 3. 8), qn sn−1 + qn bn = − Qn ∞ ck sk , k=n Some Topics in Summability Theory 39 where ck = 1 1 − , k = 0, 1, 2, .

We now have the following. 9 (Inclusion theorem) Given the methods (M, λn ) and (M, μn ), (M, λn ) ⊆ (M, μn ) if and only if ∞ ∞ |kn | < ∞ and n=0 where μ(x) = k(x) = λ(x) ∞ ∞ kn x n , λ(x) = n=0 kn = 1, n=0 ∞ λn x n , μ(x) = n=0 μn x n . n=0 Some Topics in Summability Theory 45 Proof The proof parallels that of Hardy [7, 65–68]. ∞ ∞ Let λ(x) = λn x n , μ(x) = n=0 μn x n . Both the series on the right converge for n=0 |x| < 1. Let {u n }, {vn } be the (M, λn ), (M, μn ) transforms of {sn }, respectively.

Let {tk (n)} be a sequence defined by tk (n) = 1 k k−1 sn+ν , k ∈ N . (108) ν=0 Then tk (n) is said to be the kth element of the Banach transformed sequence. If u n is said to be lim tk (n) = s, a finite number, uniformly for all n ∈ N , then k→∞ Banach summable to s. Thus if sup |tk (n) − s| → 0, for k → ∞, (109) n then we say that u n is Banach summable to s. Further if ∞ |tk (n) − tk+1 (n)| < ∞, (110) k=1 uniformly for all n ∈ N and for tk (n) as defined in (108), then the series to be absolutely Banach summable.