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The Conjugation Operator and the Hilbert Transform

  • Levan Zhizhiashvili
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 372)

Abstract

Throughout the book we use the notation \( T = \left[ { - \pi ,\pi } \right],\mathbb{R} = \left] { - \infty , + \infty } \right[, \) and \( \mathbb{R}_ + = \left[ {0, + \infty } \right[. \) Given \( p \in \left] {0, + \infty } \right[,L^p \left( T \right) \) will stand for the set of all 2π-periodic measurable functions f:ℝ→ℝ for which the expression
$$ \left\| f \right\|_p = \left\{ {\frac{1} {{2\pi }}\int\limits_T {\left| {f\left( x \right)} \right|^p dx} } \right\}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 p}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$p$}}} $$
is finite, while for p = ∞, we will assume
$$ L^\infty \left( T \right) = C\left( T \right),\left\| f \right\|_\infty = \mathop {\sup }\limits_{x \in T} \left| {f\left( x \right)} \right|. $$

Keywords

Orlicz Space Trigonometric Series Conjugation Operator Hilbert Transform Trigonometric Fourier Series 
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

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Levan Zhizhiashvili
    • 1
  1. 1.Department of Mechanics and MathematicsTbilisi State UniversityTbilisiUSA

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