Hilbert transformation and representation of the ax + b group
Pei Dang1; Hua Liu2; Tao Qian3
Source PublicationCanadian Mathematical Bulletin

In this paper we study the Hilbert transformations over L(R) and L(T) from the view point of symmetry. For a linear operator over L(R) commutative with the ax + b group, we show that the operator is of the form λI+nH, where I and H are the identity operator and Hilbert transformation, respectively, and λ, n are complex numbers. In the related literature this result was proved by first invoking the boundedness result of the operator using some machinery. In our setting the boundedness is a consequence of the boundedness of the Hilbert transformation. The methodology that we use is the Gelfand-Naimark representation of the ax + b group. Furthermore, we prove a similar result on the unit circle. Although there does not exist a group like the ax + b group on the unit circle, we construct a semigroup that plays the same symmetry role for the Hilbert transformations over the circle L(T).

KeywordHilbert Transform Singular Integral The Ax + b Group
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Indexed BySCIE
WOS Research AreaMathematics
WOS SubjectMathematics
WOS IDWOS:000426536300005
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Document TypeJournal article
CollectionUniversity of Macau
Affiliation1.Faculty of Information Technology, Macau University of Science and Technology, Macau, China
2.Department of Mathematics, Tianjin University of Technology and Education, Tianjin 300222, China
3.Department of Mathematics, University of Macau, Macau, China
First Author AffilicationUniversity of Macau
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GB/T 7714
Pei Dang,Hua Liu,Tao Qian. Hilbert transformation and representation of the ax + b group[J]. Canadian Mathematical Bulletin,2018,61(1):70-84.
APA Pei Dang,Hua Liu,&Tao Qian.(2018).Hilbert transformation and representation of the ax + b group.Canadian Mathematical Bulletin,61(1),70-84.
MLA Pei Dang,et al."Hilbert transformation and representation of the ax + b group".Canadian Mathematical Bulletin 61.1(2018):70-84.
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