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

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1. CMB Online first

Li, Junfeng; Yu, Haixia
Oscillatory Hyper-Hilbert Transform Associated with Plane Curves
In this paper, the bounded properties of oscillatory hyper-Hilbert transform along certain plane curves $\gamma(t)$ $$T_{\alpha,\beta}f(x,y)=\int_{0}^1f(x-t,y-\gamma(t))e^{ i t^{-\beta}}\frac{\textrm{d}t}{t^{1+\alpha}}$$ were studied. For a general curves, these operators are bounded in ${L^2(\mathbb{R}^{2})}$, if $\beta\geq 3\alpha$. And their boundedness in $L^p(\mathbb{R}^{2})$ were also obtained, whenever $\beta\gt 3\alpha$, $\frac{2\beta}{2\beta-3\alpha}\lt p\lt \frac{2\beta}{3\alpha}$.

Keywords:oscillatory hyper-Hilbert transform, oscillatory integral
Categories:42B20, 42B35

2. CMB 2017 (vol 61 pp. 70)

Dang, Pei; Liu, Hua; Qian, Tao
Hilbert Transformation and Representation of the $ax+b$ Group
In this paper we study the Hilbert transformations over $L^2(\mathbb{R})$ and $L^2(\mathbb{T})$ from the viewpoint of symmetry. For a linear operator over $L^2(\mathbb{R})$ commutative with the ax+b group we show that the operator is of the form $ \lambda I+\eta H, $ where $I$ and $H$ are the identity operator and Hilbert transformation respectively, and $\lambda,\eta$ are complex numbers. In the related literature this result was proved through first invoking the boundedness result of the operator, proved though a big machinery. In our setting the boundedness is a consequence of the boundedness of the Hilbert transformation. The methodology that we use is Gelfand-Naimark's representation of the ax+b group. Furthermore we prove a similar result on the unit circle. Although there does not exist a group like ax+b on the unit circle, we construct a semigroup to play the same symmetry role for the Hilbert transformations over the circle $L^2(\mathbb{T}).$

Keywords:singular integral, Hilbert transform, the $ax+b$ group
Categories:30E25, 44A15, 42A50

3. CMB 2016 (vol 59 pp. 497)

De Carli, Laura; Samad, Gohin Shaikh
One-parameter Groups of Operators and Discrete Hilbert Transforms
We show that the discrete Hilbert transform and the discrete Kak-Hilbert transform are infinitesimal generator of one-parameter groups of operators in $\ell^2$.

Keywords:discrete Hilbert transform, groups of operators, isometries
Categories:42A45, 42A50, 41A44

4. CMB 2006 (vol 49 pp. 203)

Çömez, Doğan
The Ergodic Hilbert Transform for Admissible Processes
It is shown that the ergodic Hilbert transform exists for a class of bounded symmetric admissible processes relative to invertible measure preserving transformations. This generalizes the well-known result on the existence of the ergodic Hilbert transform.

Keywords:Hilbert transform, admissible processes
Categories:28D05, 37A99

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