《Summation of The Fourier Series of Orthogonal Functions》求取 ⇩

INTRODUCTION1

CHAPTER Ⅰ．Normal orthogonal system of functions5

1．Convergence and summability (C,1) of the series of orthogonal functions5

2．Riesz-summability of the series of orthogonal functions13

3．Lebesgue＇s functions of a normal orthogonal system16

4．Conditions for the completeness of the systems of orthogonal functions23

5．An extension of Parseval＇s formula28

CHAPTER Ⅱ．The trigonometrical series33

1．Cesàro summability of the Fourier series of f(x) and the mean functions of f(x)33

2．Convergence problem．Criterions for the convergence of Fourier series51

3．Convergence of the allied series of a Fourier series77

4．Cesàro-summability of Fourier series of functions belonging to Lipschitz class81

5．Summability of the derived series of a Fourier series82

CHAPTER Ⅲ．Absolute convergence of Fourier series87

1．The class of functions with absolutely convergent Fourier series87

2．Absolute convergence of a Fourier series at a given point90

3．Absolute convergence of the Fourier series of functions bounded variation96

4．A necssary condition for absolute convergence97

CHAPTER Ⅳ．Absolute Cesàro summability of positive order for a Fourier series99

1．Functions of bounded variation99

2．A generalization of Hardy＇s theorem with an application to the absolute summability of Fourier series104

CHAPTER Ⅴ．Absolute Cesàro summability of negative order for a Fourier series at a given point111

1．Lemmae113

2．Summability of power series118

3．Criteria for the summability |C,α|,α<0122

4．An extension of a theorem of Zygmund123

5．Further criteria for the summability |C,α|,α<0124

CHAPTER Ⅵ．Absolute convergence of the allied series of a Fourier series133

1．Introduction133

2．The function zβ (w)136

3．Lemmas concerning series and fractional integrals141

4．The Fourier series of odd functions of bounded variation143

5．The function |t|p [|t|-pψ(t)]-α under the conditions143

6．The function |t|p[|t|-pψ(t)]-α and the series ？ Bn(x)146

7．Importance of the condition X(t) t-1 ？ L in Theorem 3150

8．Summability |C,α|,α<0 of the allied series155

9．Extension of Bosanquet＇s criterion to the summability of negative order157

CHAPTER Ⅶ．Laplace＇s series of hyperspherical function159

1．The summability (C,k),k？p-2 for the series163

2．The summability (C,k),k>(p-2)/2 for the series167

3．The order p-2 is critical169

BIBLIOGRAPHY173