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
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