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Search: All articles in the CJM digital archive with keyword Hilbert transform

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1. CJM 2011 (vol 63 pp. 1161)

Neuwirth, Stefan; Ricard, Éric
Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group
We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group $\varGamma$ and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum $\varLambda\subseteq\varGamma$, the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1.

Keywords:Fourier multiplier, Toeplitz Schur multiplier, lacunary set, unconditional approximation property, Hilbert transform, Riesz projection
Categories:47B49, 43A22, 43A46, 46B28

2. CJM 2003 (vol 55 pp. 1231)

Havin, Victor; Mashreghi, Javad
Admissible Majorants for Model Subspaces of $H^2$, Part I: Slow Winding of the Generating Inner Function
A model subspace $K_\Theta$ of the Hardy space $H^2 = H^2 (\mathbb{C}_+)$ for the upper half plane $\mathbb{C}_+$ is $H^2(\mathbb{C}_+) \ominus \Theta H^2(\mathbb{C}_+)$ where $\Theta$ is an inner function in $\mathbb{C}_+$. A function $\omega \colon \mathbb{R}\mapsto[0,\infty)$ is called an admissible majorant for $K_\Theta$ if there exists an $f \in K_\Theta$, $f \not\equiv 0$, $|f(x)|\leq \omega(x)$ almost everywhere on $\mathbb{R}$. For some (mainly meromorphic) $\Theta$'s some parts of Adm $\Theta$ (the set of all admissible majorants for $K_\Theta$) are explicitly described. These descriptions depend on the rate of growth of $\arg \Theta$ along $\mathbb{R}$. This paper is about slowly growing arguments (slower than $x$). Our results exhibit the dependence of Adm $B$ on the geometry of the zeros of the Blaschke product $B$. A complete description of Adm $B$ is obtained for $B$'s with purely imaginary (``vertical'') zeros. We show that in this case a unique minimal admissible majorant exists.

Keywords:Hardy space, inner function, shift operator, model, subspace, Hilbert transform, admissible majorant
Categories:30D55, 47A15

3. CJM 2003 (vol 55 pp. 1264)

Havin, Victor; Mashreghi, Javad
Admissible Majorants for Model Subspaces of $H^2$, Part II: Fast Winding of the Generating Inner Function
This paper is a continuation of Part I [6]. We consider the model subspaces $K_\Theta=H^2\ominus\Theta H^2$ of the Hardy space $H^2$ generated by an inner function $\Theta$ in the upper half plane. Our main object is the class of admissible majorants for $K_\Theta$, denoted by Adm $\Theta$ and consisting of all functions $\omega$ defined on $\mathbb{R}$ such that there exists an $f \ne 0$, $f \in K_\Theta$ satisfying $|f(x)|\leq\omega(x)$ almost everywhere on $\mathbb{R}$. Firstly, using some simple Hilbert transform techniques, we obtain a general multiplier theorem applicable to any $K_\Theta$ generated by a meromorphic inner function. In contrast with [6], we consider the generating functions $\Theta$ such that the unit vector $\Theta(x)$ winds up fast as $x$ grows from $-\infty$ to $\infty$. In particular, we consider $\Theta=B$ where $B$ is a Blaschke product with ``horizontal'' zeros, i.e., almost uniformly distributed in a strip parallel to and separated from $\mathbb{R}$. It is shown, among other things, that for any such $B$, any even $\omega$ decreasing on $(0,\infty)$ with a finite logarithmic integral is in Adm $B$ (unlike the ``vertical'' case treated in [6]), thus generalizing (with a new proof) a classical result related to Adm $\exp(i\sigma z)$, $\sigma>0$. Some oscillating $\omega$'s in Adm $B$ are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm $\exp(i\sigma z)$, $\sigma>0$, and to de Branges' space $\mathcal{H}(E)$.

Keywords:Hardy space, inner function, shift operator, model, subspace, Hilbert transform, admissible majorant
Categories:30D55, 47A15

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