Fourier Series with Respect to General Orthogonal Systems [Paperback]

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 {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. Garsias 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.Springer Book Archives