Abstract
在此篇論文中,我們所要證明的是:如果.absolute.(f )是在(0,1)這個開區間中的一個可積分函數,a(k)為f之 Walsh-Fourier係數,而且對於某個正整數m來講,a(k)滿足 lim limsup .SIGMA.([.lambda.*n]-k+1)*.absolute. (.DELTA.(m,a(k))))/([.lambda.*n]-n)=0;在這裡的lim是取.lambda.遞減至1,limsup 是取n趨近於正無窮大,而 .SIGMA.則是將k由n加到[.lambda.*n];[.]表示高斯符號, .DELTA.(m,a(k))則是表示a(k)的m次差分.則f的Walsh- Fourier級數的部分和S(n;f;x)在幾乎每一點上均會收斂至f(x).除此之外,在滿足相同的條件之下,對於所有的r落在0和1/m之間,當n趨近於正無窮大時,我們可以得到.int. (.absolute.(S(n;f;x)-f(x)))^rdx會趨近於0,這裡的積分範圍是由0積至1.這些結果均推廣了Moricz的定理.而為了推廣這個結果,在這篇論文當中,我們亦推廣了Fine的兩個結果,Fine的這兩個結果是將Walsh-Dirichlet核以及 Walsh-Fejer核做了適當的展開與估計.本論文共分為五節:第一節為前言,定義Walsh函數並且陳述他們的一些性質;第二節為主要結果,敘述主要定理及其一些應用;為了證明主要定理,我們必須先推廣Fine的兩個結論,詳細的敘述請讀者參見第三節輔助定理;第四節則為主要定理之證明;第五節則將Fine的其中一個結果寫成了更好的形式.In this paper, we prove that if .absolute.(f) is integrable inthe open interval (0,1), a(k) are the Walsh-Fouriercoefficients of f and for some positive interger m, a(k)satisfy the condition: lim limsup .SIGMA.([.lambda.*n]-k+1)*.absolute.( .DELTA.(m,a(k))))/([.lambda.*n]-n)=0; Here, lim istaken .lambda. decreases to 1, limsup is taken n increases to.inf., and .SIGMA. is the summation of k from n to [.lambda.*n]; [.lambda.*n] means the integer part of .lambda.*n, .DELTA.(m,a(k)) means the mth difference of a(k). Then the partial sumsS(n;f;x) of the Walsh-Fourier series of f converge to f(x)almost everywhere, Moreover, .int.(.absolute.(S(n;f;x)-f(x)))^rdx approaches to 0 as n approaches to .inf. for 0<r<1/m,where the domain of the integral is taken from 0 to 1. Thisextends the result of Moricz. In this paper, we also extend tworesults of Fine, this two results give certain expansions andestimates of the Walsh -Dirichlet kernels and the Walsh-Fejerkernels. This paper has five sections: Section 1 is theintroduction, we state the definition and some properties ofthe Walsh functions; Section 2 is the main result, we state themain theorem and its applications; Before proving the maintheorem, we have to extend two results of Fine, for a detail,we refer the readers to Section 3: the auxiliary results;Section 4 is the proof of the main theorem; Section 5 is theconcluding remark.