Ebook: Trigonometric Fourier Series and Their Conjugates

This book presents in a coherent way the results obtained in the following aspects of the theory of multiple trigonometric Fourier series: the existence and properties of the conjugates and Hilbert transforms of integrable functions of several variables; convergence of Fourier series and their conjugates, as well as their summability by Ces� ro and Abel-Poisson methods; and approximating properties of Ces� ro means of Fourier series and their conjugates.
Special emphasis is put on new effects which arise from dealing with multiple series and which are not inherent in the one-dimensional case. Unsolved problems are formulated separately.
Audience: This volume will prove useful to both graduate students and research workers in the field of Fourier analysis, approximations and expansions, integral transforms, and operational calculus.

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This book presents in a coherent way the results obtained in the following aspects of the theory of multiple trigonometric Fourier series: the existence and properties of the conjugates and Hilbert transforms of integrable functions of several variables; convergence of Fourier series and their conjugates, as well as their summability by Ces`ro and Abel-Poisson methods; and approximating properties of Ces`ro means of Fourier series and their conjugates.
Special emphasis is put on new effects which arise from dealing with multiple series and which are not inherent in the one-dimensional case. Unsolved problems are formulated separately.
Audience: This volume will prove useful to both graduate students and research workers in the field of Fourier analysis, approximations and expansions, integral transforms, and operational calculus.

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This book presents in a coherent way the results obtained in the following aspects of the theory of multiple trigonometric Fourier series: the existence and properties of the conjugates and Hilbert transforms of integrable functions of several variables; convergence of Fourier series and their conjugates, as well as their summability by Ces`ro and Abel-Poisson methods; and approximating properties of Ces`ro means of Fourier series and their conjugates.
Special emphasis is put on new effects which arise from dealing with multiple series and which are not inherent in the one-dimensional case. Unsolved problems are formulated separately.
Audience: This volume will prove useful to both graduate students and research workers in the field of Fourier analysis, approximations and expansions, integral transforms, and operational calculus.
Content:
Front Matter....Pages i-xii
Front Matter....Pages 1-1
The Conjugation Operator and the Hilbert Transform....Pages 3-33
Pointwise Convergence and Summability of Trigonometric Series....Pages 35-70
Convergence and Summability of Trigonometric Fourier Series and Their Conjugates in the Spaces $L^p left( T right),p in left] {0, + infty } right[$ ....Pages 71-92
Some Approximating Properties of Ces`ro Means of the Series $sigma left[ f right]$ and $bar sigma left[ f right]$ ....Pages 93-116
Front Matter....Pages 117-117
Conjugate Functions and Hilbert Transforms of Functions of Several Variables....Pages 119-166
Convergence and Summability at a Point or Almost Everywhere of Multiple Trigonometric Fourier Series and Their Conjugates....Pages 167-203
Some Approximating Properties of n-Fold Ces`ro Means of the Series $sigma _n left[ {text{f}} right]$ and $bar sigma _n left[ {{text{f,B}}} right]$ ....Pages 205-224
Convergence and Summability of Multiple Trigonometric Fourier Series and Their Conjugates in the Spaces LP (Tn), p?]o,+?]....Pages 225-242
Summability of Series $sigma _2 left[ {text{f}} right]$ and $bar sigma _2 left[ {{text{f,B}}} right]$ by a Method of the Marcinkiewicz Type....Pages 243-266
Back Matter....Pages 267-308

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This book presents in a coherent way the results obtained in the following aspects of the theory of multiple trigonometric Fourier series: the existence and properties of the conjugates and Hilbert transforms of integrable functions of several variables; convergence of Fourier series and their conjugates, as well as their summability by Ces`ro and Abel-Poisson methods; and approximating properties of Ces`ro means of Fourier series and their conjugates.
Special emphasis is put on new effects which arise from dealing with multiple series and which are not inherent in the one-dimensional case. Unsolved problems are formulated separately.
Audience: This volume will prove useful to both graduate students and research workers in the field of Fourier analysis, approximations and expansions, integral transforms, and operational calculus.
Content:
Front Matter....Pages i-xii
Front Matter....Pages 1-1
The Conjugation Operator and the Hilbert Transform....Pages 3-33
Pointwise Convergence and Summability of Trigonometric Series....Pages 35-70
Convergence and Summability of Trigonometric Fourier Series and Their Conjugates in the Spaces $L^p left( T right),p in left] {0, + infty } right[$ ....Pages 71-92
Some Approximating Properties of Ces`ro Means of the Series $sigma left[ f right]$ and $bar sigma left[ f right]$ ....Pages 93-116
Front Matter....Pages 117-117
Conjugate Functions and Hilbert Transforms of Functions of Several Variables....Pages 119-166
Convergence and Summability at a Point or Almost Everywhere of Multiple Trigonometric Fourier Series and Their Conjugates....Pages 167-203
Some Approximating Properties of n-Fold Ces`ro Means of the Series $sigma _n left[ {text{f}} right]$ and $bar sigma _n left[ {{text{f,B}}} right]$ ....Pages 205-224
Convergence and Summability of Multiple Trigonometric Fourier Series and Their Conjugates in the Spaces LP (Tn), p?]o,+?]....Pages 225-242
Summability of Series $sigma _2 left[ {text{f}} right]$ and $bar sigma _2 left[ {{text{f,B}}} right]$ by a Method of the Marcinkiewicz Type....Pages 243-266
Back Matter....Pages 267-308
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