The fundamental problem of the theory of Fourier series consists of the investigation of the connections between the metric properties of the function expanded, the behavior of its Fourier coefficients {cn} with respect to an ortho normal system of functions {
The fundamental problem of the theory of Fourier series consists of the investigation of the connections between the metric properties of the function expanded, the behavior of its Fourier coefficients {cn} with respect to an ortho normal system of functions {
Produktdetails
Produktdetails
Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge .86
Terminology. Preliminary Information.- I. Convergence of Fourier Series in the Classical Sense. Lebesgue Functions of Bounded Systems.- 1. The Fundamental Inequality.- 2. The Logarithmic Growth of the Lebesgue Functions. Divergence of Fourier Series.- 3. Series with Decreasing Coefficients.- 4. Generalizations, Counterexamples, Problems.- 5. The Stability of the Orthogonalization Operator.- II. Convergence Almost Everywhere; Conditions on the Coefficients.- 1. The Class S?.- 2. Garsia's Theorem.- 3. The Coefficients of Convergent Series in Complete Systems.- 4. Extension of a System of Functions to an ONS.- III. Properties of Complete Systems; the Role of the Haar System.- 1. The Basic Construction.- 2. Divergent Fourier Series.- 3. Bases in Function Spaces and Majorants of Fourier Series.- 4. Fourier Coefficients of Continuous Functions.- 5. Some More Results about the Haar System.- IV. Series from L2 and Peculiarities of Fourier Series from the Spaces Lp.- 1. The Matrices Ak.- 2. Lebesgue Functions and Convergence Almost Everywhere.- 3. Convergence of Fourier Series of Functions from Various Classes.- 4. Sums of Fourier Series.- 5. Conditional Bases in Hubert Space.
Terminology. Preliminary Information.- I. Convergence of Fourier Series in the Classical Sense. Lebesgue Functions of Bounded Systems.- 1. The Fundamental Inequality.- 2. The Logarithmic Growth of the Lebesgue Functions. Divergence of Fourier Series.- 3. Series with Decreasing Coefficients.- 4. Generalizations, Counterexamples, Problems.- 5. The Stability of the Orthogonalization Operator.- II. Convergence Almost Everywhere; Conditions on the Coefficients.- 1. The Class S?.- 2. Garsia's Theorem.- 3. The Coefficients of Convergent Series in Complete Systems.- 4. Extension of a System of Functions to an ONS.- III. Properties of Complete Systems; the Role of the Haar System.- 1. The Basic Construction.- 2. Divergent Fourier Series.- 3. Bases in Function Spaces and Majorants of Fourier Series.- 4. Fourier Coefficients of Continuous Functions.- 5. Some More Results about the Haar System.- IV. Series from L2 and Peculiarities of Fourier Series from the Spaces Lp.- 1. The Matrices Ak.- 2. Lebesgue Functions and Convergence Almost Everywhere.- 3. Convergence of Fourier Series of Functions from Various Classes.- 4. Sums of Fourier Series.- 5. Conditional Bases in Hubert Space.
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