數列空間的矩陣變換與Tauber型定理

Matrix maps of sequence spaces and Tauberian theorems

張其棟(Chi-Tung Chang)

指導教授 ： 黃明傑 ； 陳璋泡

國立清華大學/理學院/數學系/博士(2009年)

摘要

我們首先回顧古典數列空間的矩陣變換，其相關的理論已經為人所熟知。於本篇論文中，此理論將推擴至統計收斂的概念。我們不僅特徵化統計收斂數列空間的矩陣變換，給出變換後的極限公式，並計算出相對應變換的範數。接著我們考慮其逆向的問題，亦即找出所謂的Tauber型條件，使得在此條件之下，二重數列的收斂性可以由其加權平均的統計收斂性得到。其中我們定義了新型緩速震盪與緩速遞減的條件，並發現為人所熟知的Hardy型條件、Landau型條件和Schmidt型條件都可以為Tauber型條件。最後，以上與二重數列平行的理論亦建立於多變數可測函數的角度。我們於本篇論文的結論推擴了許多近代知名的結果。

關鍵字

統計收斂 ； 矩陣變換 ； Tauber型定理

並列摘要

We first review the characterization of the matrix maps between classical sequence spaces. Such a theory has been developed for a long time. We extend this theory to the concept of A-statistical convergence. We not only give the characterization of matrix maps of A-statistically convergent sequence spaces, but also obtain the corresponding limit formula. In addition, we evaluate the operator semi-norm of such matrix maps. Next, the converse problem is concerned. More precisely, we find conditions, called Tauberian conditions, under which the ordinary convergence of double sequences follows from the statistical convergence of its weighted means. To this aim, a new type of slow oscillation and slow decrease conditions are introduced. As a result, we conclude that Hardy’s two-sided condition, Landau’s condition and Schmidt’s condition are the desired Tauberian conditions. Finally, a parallel theory to the one given for double sequences is established for measurable functions of several variables. Our results in this thesis generalize many well-known recent works.

並列關鍵字

statistical convergence ； matrix map ； Tauberian theorem ； summability

參考文獻

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