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Fourier Series

  • R. M. Dudley
  • R. Norvaiša
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

Let \( T^{1} :=\{z: |z|=1\}=\{{\rm e}^{{\rm i}\theta}: 0\leq \theta < 2\pi\} \) be the unit circle in the complex plane\(\mathbb{C}\). On \(T^{1}\) let \(d\mu = d\theta/2\pi\) be the rotationally invariant probability measure. For\(\mathbb{Z} = \{0,\pm1,\pm2, . . . \}\), the functions \(z \mapsto z^n\) or\(\theta \mapsto e^{{in}\theta}\) for \(n \in \mathbb{Z}\) form an orthonormal basis of complex \(L^2(T^1, \mu)\) (e.g. [53], Proposition 7.4.2).

Keywords

Fourier Series Uniform Convergence Invariant Probability Measure Young Inequality Complementary Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Institute of Mathematics and InformaticsVilniusLithuania

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