Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hilbert Transformation and Representation of 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 $L^2(\mathbb{T}).$

27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
16
The categories of this paper are: stat
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex