Continuity of the Bessel wavelet transform on certain Beurling-type function spaces
© Prasad et al.; licensee Springer 2013
Received: 17 November 2012
Accepted: 8 January 2013
Published: 22 January 2013
In this paper, continuity of the Bessel wavelet transform of a suitable function ϕ in terms of an appropriate mother wavelet ψ is investigated on certain Beurling-type function spaces.
Bessel wavelet transforms have applications in the study of boundary value problems on the half-line. The Hankel transform and pseudo-differential operators associated with the Bessel operator have been studied on some Beurling-type function spaces by [1, 2] and  respectively. The aim of the present paper is to study the continuity of the Bessel wavelet transform on certain Beurling-type function spaces.
2 The space
for some real l and .
The class of all such ω functions is denoted by M.
In what follows, we study the Bessel wavelet transform of infinite order on . For this purpose, we define the symbol class .
where c is a constant and denotes the Hankel transform of the basic wavelet ψ.
In this section, certain spaces of functions of Beurling-type are introduced on which Bessel wavelet transforms can be defined. First, we recall the definition of the Zemanian space .
, is denoted by , where .
Theorem 2.1 The Bessel wavelet transform is a continuous linear map of into for .
Hence, . □
3 The space
where are constants depending only on μ.
The family of seminorms generates the topology of .
is denoted by , where .
Theorem 3.1 The Bessel wavelet transform is a continuous linear map of into for .
This completes the proof. □
This work is supported by Indian School of Mines, Dhanbad, under grant No. 613002 /ISM JRF/Acad/2009.
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