MATRIX TRANSFORMATIONS FROM ABSOLUTELY CONVERGENT SERIES TO CONVERGENT SEQUENCES AS GENERAL WEIGHTED MEAN SUMMABILITY METHODS

We prove the necessary and sufficient conditions for an infinity matrix to be a mapping, from absolutely convergent series to convergent sequences, which is treated as general weighted mean summability methods. The results include a classical result by Hardy and another by Moricz and Rhoades as particular cases.

It is clear that summable (C, 1) is a special case of summableN, where p k = 1, k= 0, 1, 2,... . (1.6) Based on the above idea, Moricz and Rhoades [2] established a result for a broad class of summability methods, which include the method of summability (C, 1) as a particular case. Theorem 1.2. LetN be a weighted mean matrix determined by a sequence (p n ) of positive numbers such that the following conditions are satisfied: with the agreement that converges to the same limit L.
In this paper, we will study the matrix transformations from the space of absolutely convergent series of complex numbers, l 1 , to the space of convergent sequences of complex numbers, c. Then we shall establish a more general result for a broader class of weighted mean methods, which includes the method of summableN as a particular case if the series (1.1) is absolutely convergent.

2.
Matrix transformations from l 1 to c. Let A = (a nk ) be an infinity matrix with complex entries and let l denote the linear space of complex number sequences. For a sequence x = (x n ) ∈ l, Ax is in l and its entries are given by a nk x k , n= 0, 1, 2,... (2.1) provided the series converges to a finite complex number. The following result is well known (see [3,4]); we list it as a proposition.
converges to a finite complex number, then the linear functional f a defined on l 1 by is a continuous (bounded) linear functional on l 1 , such that The following result has been proved in [4] by using functional analysis techniques. It is also proved by summability methods. We list the following theorem without proof.

Theorem 2.3. Let A = (a nk ) be an infinity matrix with complex entries. Then
A is a mapping from l 1 to c, if and only if the following conditions are satisfied: (i) for every fixed k = 0, 1, 2,..., the sequence (a nk ) converges to a finite limit as n → ∞, (2.8) The following corollary follows from Theorem 2.3 and (2.8).

Corollary 2.4. Let A = (a nk ) be an infinity matrix with complex entries. If A is a mapping from l 1 to c, then the linear operator A is continuous (bounded) linear operator such that
(2.9) 3. Applications to summable (C, 1) and summableN. The following corollary comes immediately from Theorem 2.3, which describes an equivalent reformulation of summability by more general weighted mean methods which are matrix transformations. converges to a finite limit as n → ∞.
To generalize Theorem 1.2, we shall construct two weighted mean matrices according to the summability (3.7) and the following summability method:   In a particular case, as mentioned by Moricz and Rhoades [2], taking p k = 1, for k = 0, 1, 2,..., we find the Hardy's result, Theorem 1.1, if that the series (1.1) is absolutely convergent, that is, for any x = (x n ) ∈ l 1 ,

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions. However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models. In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset. Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning • Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation