Fourier Series

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


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).


Fourier Series Uniform Convergence Invariant Probability Measure Young Inequality Complementary Function 
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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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