Tauberian theorems for (N̄, p, q, w) summable triple sequences of fuzzy numbers

In this paper, we determine necessary and sufficient Tauberian conditions under which convergence in Pringsheim's sense of a double sequence of fuzzy numbers follows from its $(C,1,1)$ summability. These conditions are satisfied if the double sequence of fuzzy numbers is slowly oscillating in different senses. We also construct some interesting double sequences of fuzzy numbers.

After a brief summary of Tauberian conditions for ordinary sequences of numbers, we consider summability of double sequences of real or complex numbers by weighted mean methods which are not necessarily products of related weighted mean methods in one variable. Our goal is to obtain Tauberian conditions under which convergence of a double sequence follows from its summability, where convergence is understood in Pringsheim′s sense. In the case of double sequences of real numbers, we present necessary and sufficient Tauberian conditions, which are so‐called one‐sided conditions. Corollaries allow these Tauberian conditions to be replaced by Schmidt‐type slow decrease conditions. For double sequences of complex numbers, we present necessary and sufficient so‐called two‐sided Tauberian conditions. In particular, these conditions are satisfied if the summable double sequence is slowly oscillating.

The first named author has recently proved necessary and sufficient Tauberian conditions under which statistical convergence (introduced by H. Fast in 1951) follows from statistical summability (C, 1). The aim of the present paper is to generalize these results to a large class of summability methods (,p) by weighted means. Let p = (pk : k = 0,1, 2,...) be a sequence of nonnegative numbers such that po > 0 and Let (xk) be a sequence of real or complex numbers and set for n = 0,1, 2,.... We present necessary and sufficient conditions under which the existence of the limit st-lim xk = L follows from that of st-lim tn = L, where L is a finite number. If (xk) is a sequence of real numbers, then these are one-sided Tauberian conditions. If (xk) is a sequence of complex numbers, then these are two-sided Tauberian conditions.

In this paper, we define the weighted mean method $$(\overline{N},p,q)$$ of double sequences of fuzzy numbers and give necessary and sufficient Tauberian conditions under which convergence in Pringsheim’s sense of a double sequence of fuzzy numbers follows from its $$(\overline{N},p,q)$$ summability. These conditions are weaker than the weighted analogues of Landau’s conditions and Schmidt’s slow oscillation condition in some senses for two-dimensional case.

In this paper, we present statistical weighted mean ( briefly, (N?,p,q,1,1)) summability method for double sequences of fuzzy numbers and give necessary and sufficient Tauberian conditions under which statistical convergence of a double sequence of fuzzy numbers follows from its statistical (N?,p,q,1,1) summability. Furthermore, we apply our new method of summability to prove a fuzzy Korovkin type approximation theorem for a double sequence of fuzzy positive linear operators.

Let ( u m n s ) ${(u_{mns})}$ be a (C,1,1,1) summable triple sequence of real numbers. We give one-sided Tauberian conditions of Landau and Hardy type under which ( u m n s ) ${(u_{mns})}$ converges in Pringsheim's sense. We prove that ( u m n s ) ${(u_{mns})}$ converges in Pringsheim's sense if ( u m n s ) ${(u_{mns})}$ is slowly oscillating in certain senses. Moreover, we extend a Tauberian theorem given by Móricz [Studia Math. 110 (1994), 83–96] for double sequences to triple sequences.

A sequence $$(x_n)$$ of fuzzy numbers is said to be summable to a fuzzy number L by the logarithmic mean method $$(\ell ,2)$$ if $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\ell _n^{(2)}}\sum _{k=1}^{n}\frac{x_k}{k\ell _k}=L \end{aligned}$$where $$\begin{aligned} \ell _n^{(2)}=\sum _{k=1}^{n}\frac{1}{k\ell _k}\sim \log (\log n). \end{aligned}$$We prove that the ordinary convergence of $$(x_n)$$ implies its $$(\ell ,2)$$ summability. The converse implication is not necessarily true. Namely, the $$(\ell ,2)$$ summability of $$(x_n)$$ may not imply the convergence of $$(x_n)$$. However, under certain additional conditions the converse may hold. Such conditions are called Tauberian conditions, and the resulting theorem is said to be a Tauberian theorem. In this paper, we provide necessary and sufficient Tauberian conditions to transform $$(\ell ,2)$$ summable sequences of fuzzy numbers into convergent sequences of fuzzy numbers with preserving the limit.

In this paper, we have determined necessary and sufficient Tauberian conditions under which statistically convergence follows from statistically (C,1)-convergence of sequences of fuzzy numbers. Our conditions are satisfied if a sequence of fuzzy numbers is statistically slowly oscillating. Also, under additional conditions it is proved that a bounded sequence of fuzzy numbers which is (C,1)-level-convergent to its statistical limit superior is statistically convergent.

On the Cesàro convergence of sequences of fuzzy numbers

We have recently proved Tauberian conditions (for single sequences) relating to statistical convergence following statistical summability by weighted means in non-archimedean fields. In this paper, statistically summable double sequences by weighted means for which the necessary and sufficient Tauberian conditions over non-archimedean fields are discussed.

Let x = (xmn) be a double sequence of real or complex numbers. The Ar,δ-transform of a sequence (xmn) is defined by (Ar,δx)mn=σmnr,δ(x)=1(m+1)(n+1)∑j=0m∑k=0n(1+rj)(1+δk)xjk, 0<r,δ<1We say that (xmn) a sequence is (Ar,δ,1,1) summable to l if (σmnr,δ(x)) has a finite limit l. It is known that if limm,n→∞xmn=l and (xmn) is bounded, then the limit limm,n→∞σmnr,δ(x)=l exists. But the inverse of this implication is not true in general. Our aim is to obtain necessary and sufficient conditions for (Ar,δ,1,1) summability method under which the inverse of this implication holds. Following Tauberian theorems for (Ar,δ, 1, 1) summability method, we also define Ar and Aδ transformations of double sequences and obtain Tauberian theorems for the (Ar,δ, 1, 0) and (Ar,δ, 0, 1) summabillity methods.

In this paper, we obtain necessary and sufficient conditions, under which convergence of a double sequence in Pringsheim's sense follows from its weighted-Cesaro summability. These Tauberian conditions are one-sided or two-sided if it is a sequence of real or complex numbers, respectively.

The uniformity level of simulation signal is expressed as a fuzzy number by using the fitting method, and its contrast analysis is naturally ascribed to the sequence of fuzzy numbers. Considering the influence of information quantity upon sequence of fuzzy numbers, a sequencing method which ponders the generalized average value, deviation, and information quantity of fuzzy number comprehensively is proposed. Then, a contrast analysis method of uniformity of simulation signal based on the sequence of fuzzy numbers is presented. It is shown by the application of the method in simulation system of target acoustic signal that, the analysis result can provide auxiliary decision-making support for the adjustment and test of signal simulation system.

We define weighted mean summability method of double sequences in intuitionistic fuzzy normed spaces(IFNS), and obtain necessary and sufficient Tauberian conditions under which convergence of double sequences in IFNS follows from their weighted mean summability. This study reveals also Tauberian results for some known summation methods in the special cases.

Let $s: [1, \infty) \to \C$ be a locally integrable function in Lebesgue's sense on the infinite interval $[1, \infty)$. We say that $s$ is summable $(L, 1)$ if there exists some $A\in \C$ such that $$\lim_{t\to \infty} \tau(t) = A, \quad {\rm where} \quad \tau(t):= {1\over \log t} \int^t_1 {s(u) \over u} du.\leqno(*)$$ It is clear that if the ordinary limit $s(t) \to A$ exists, then the limit $\tau(t) \to A$ also exists as $t\to \infty$. We present sufficient conditions, which are also necessary in order that the converse implication hold true. As corollaries, we obtain so-called Tauberian theorems which are analogous to those known in the case of summability $(C,1)$. For example, if the function $s$ is slowly oscillating, by which we mean that for every $\e>0$ there exist $t_0 = t_0 (\e) > 1$ and $\lambda=\lambda(\e) > 1$ such that $$|s(u) - s(t)| \le \e \quad {\rm whenever}\quad t_0 \le t < u \le t^\lambda,$$ then the converse implication holds true: the ordinary convergence $\lim_{t\to \infty} s(t) = A$ follows from (*). We also present necessary and sufficient Tauberian conditions under which the ordinary convergence of a numerical sequence $(s_k)$ follows from its logarithmic summability. Among others, we give a more transparent proof of an earlier Tauberian theorem due to Kwee [3].