The Conjugation Operator and the Hilbert Transform

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


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
$$ L^\infty \left( T \right) = C\left( T \right),\left\| f \right\|_\infty = \mathop {\sup }\limits_{x \in T} \left| {f\left( x \right)} \right|. $$


Orlicz Space Trigonometric Series Conjugation Operator Hilbert Transform Trigonometric Fourier Series 
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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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