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Search: MSC category 44A15 ( Special transforms (Legendre, Hilbert, etc.) )

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1. 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

2. CMB 2004 (vol 47 pp. 389)

He, Jianxun
An Inversion Formula of the Radon Transform Transform on the Heisenberg Group
In this paper we give an inversion formula of the Radon transform on the Heisenberg group by using the wavelets defined in [3]. In addition, we characterize a space such that the inversion formula of the Radon transform holds in the weak sense.

Keywords:wavelet transform, Radon transform, Heisenberg group
Categories:43A85, 44A15

3. CMB 2000 (vol 43 pp. 472)

Oberlin, Daniel M.
An Estimate For a Restricted X-Ray Transform
This paper contains a geometric proof of an estimate for a restricted x-ray transform. The result complements one of A.~Greenleaf and A.~Seeger.


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