Fourier Series with Respect to General Orthogonal Systems

ISBN-13: 9783642660580 / Angielski / Miękka / 2011 / 138 str.

A. Olevskii; B. P. Marshall; H. J. Christoffers

Opis

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 with respect to an ortho normal system of functions {"

Spis treści

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.