Calderón's Reproducing Formula for Hankel Convolution

Calderón-type reproducing formula for Hankel convolution is established using the theory of Hankel transform.


Introduction
Calder ón's formula [3] involving convolutions related to the Fourier transform is useful in obtaining reconstruction formula for wavelet transform besides many other applications in decomposition of certain function spaces.It is expressed as follows: where φ : R n → C and φ t (x) = t −n φ(x/t), t > 0. For conditions of validity of identity (1.1), we may refer to [3].
Hankel convolution introduced by Hirschman Jr. [5] related to the Hankel transform was studied at length by Cholewinski [1] and Haimo [4].Its distributional theory was developed by Marrero and Betancor [6].Pathak and Pandey [8] used Hankel convolution in their study of pseudodifferential operators related to the Bessel operator.Pathak and Dixit [7] exploited Hankel convolution in their study of Bessel wavelet transforms.In what follows, we give definitions and results related to the Hankel convolution [5] to be used in the sequel.
Let γ be a positive real number.Set where J γ−1/2 denotes the Bessel function of order γ − 1/2.

Calder ón's formula
In this section, we obtain Calder ón's reproducing identity using the properties of Hankel transform and Hankel convolutions.

.2)
Proof.Taking Hankel transform of the right-hand side of (2.2), we get Now, by putting aξ = ω, we get (2.4) Hence, the result follows.
The equality (2.2) can be interpreted in the following L 2 -sense.