136 research outputs found

    Comparison theorems for summability methods of sequences of fuzzy numbers

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    In this study we compare Ces\`{a}ro and Euler weighted mean methods of summability of sequences of fuzzy numbers with Abel and Borel power series methods of summability of sequences of fuzzy numbers. Also some results dealing with series of fuzzy numbers are obtained.Comment: publication information is added, typos correcte

    Tauberian theorems for weighted mean statistical summability of double sequences of fuzzy numbers

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    We discuss Tauberian conditions under which the statistical convergence of double sequences of fuzzy numbers follows from the statistical convergence of their weighted means. We also prove some other results which are necessary to establish the main results

    TAUBERIAN THEOREMS FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY OF INTEGRALS

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    Let qq be a positive weight function on R+:=[0,)\mathbf{R}_{+}:=[0, \infty) which is integrable in Lebesgue's sense over every finite interval (0,x)(0,x) for 0000, Q(0)=0Q(0)=0 and Q(x)Q(x) \rightarrow \infty as xx \to \infty .Given a real or complex-valued function fLloc1(R+)f \in L^{1}_{loc} (\mathbf{R}_{+}), we define s(x):=0xf(t)dts(x):=\int_{0}^{x}f(t)dt andτq(0)(x):=s(x),τq(m)(x):=1Q(x)0xτq(m1)(t)q(t)dt(x>0,m=1,2,...),\tau^{(0)}_q(x):=s(x), \tau^{(m)}_q(x):=\frac{1}{Q(x)}\int_0^x \tau^{(m-1)}_q(t) q(t)dt\,\,\, (x>0, m=1,2,...),provided that Q(x)>0Q(x)>0. We say that 0f(x)dx\int_{0}^{\infty}f(x)dx is summable to LL by the mm-th iteration of weighted mean method determined by the function q(x)q(x), or for short, (N,q,m)(\overline{N},q,m) integrable to a finite number LL iflimxτq(m)(x)=L.\lim_{x\to \infty}\tau^{(m)}_q(x)=L.In this case, we write s(x)L(N,q,m)s(x)\rightarrow L(\overline{N},q,m). It is known thatif the limit limxs(x)=L\lim _{x \to \infty} s(x)=L exists, then limxτq(m)(x)=L\lim _{x \to \infty} \tau^{(m)}_q(x)=L also exists. However, the converse of this implicationis not always true. Some suitable conditions together with the existence of the limit limxτq(m)(x)\lim _{x \to \infty} \tau^{(m)}_q(x), which is so called Tauberian conditions, may imply convergence of limxs(x)\lim _{x \to \infty} s(x). In this paper, one- and two-sided Tauberian conditions in terms of the generating function and its generalizations for (N,q,m)(\overline{N},q,m) summable integrals of real- or complex-valued functions have been obtained. Some classical type Tauberian theorems given for Ces\`{a}ro summability (C,1)(C,1) and weighted mean method of summability (N,q)(\overline{N},q) have been extended and generalized. 
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