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INTRODUCTION 1

CHAPTER Ⅰ．Normal orthogonal system of functions 5

1．Convergence and summability (C,1) of the series of orthogonal functions 5

2．Riesz-summability of the series of orthogonal functions 13

3．Lebesgue＇s functions of a normal orthogonal system 16

4．Conditions for the completeness of the systems of orthogonal functions 23

5．An extension of Parseval＇s formula 28

CHAPTER Ⅱ．The trigonometrical series 33

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 series 51

3．Convergence of the allied series of a Fourier series 77

4．Cesàro-summability of Fourier series of functions belonging to Lipschitz class 81

5．Summability of the derived series of a Fourier series 82

CHAPTER Ⅲ．Absolute convergence of Fourier series 87

1．The class of functions with absolutely convergent Fourier series 87

2．Absolute convergence of a Fourier series at a given point 90

3．Absolute convergence of the Fourier series of functions bounded variation 96

4．A necssary condition for absolute convergence 97

CHAPTER Ⅳ．Absolute Cesàro summability of positive order for a Fourier series 99

1．Functions of bounded variation 99

2．A generalization of Hardy＇s theorem with an application to the absolute summability of Fourier series 104

CHAPTER Ⅴ．Absolute Cesàro summability of negative order for a Fourier series at a given point 111

1．Lemmae 113

2．Summability of power series 118

3．Criteria for the summability |C,α|,α<0 122

4．An extension of a theorem of Zygmund 123

5．Further criteria for the summability |C,α|,α<0 124

CHAPTER Ⅵ．Absolute convergence of the allied series of a Fourier series 133

1．Introduction 133

2．The function zβ (w) 136

3．Lemmas concerning series and fractional integrals 141

4．The Fourier series of odd functions of bounded variation 143

5．The function |t|p [|t|-pψ(t)]-α under the conditions 143

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 3 150

8．Summability |C,α|,α<0 of the allied series 155

9．Extension of Bosanquet＇s criterion to the summability of negative order 157

CHAPTER Ⅶ．Laplace＇s series of hyperspherical function 159

1．The summability (C,k),k？p-2 for the series 163

2．The summability (C,k),k>(p-2)/2 for the series 167

3．The order p-2 is critical 169

BIBLIOGRAPHY 173