On Continuous Wavelet Transform of Complex Radon Measures on ${\mathbb{R}}^d$

This paper investigates a necessary and sufficient condition for a measurable function on ${\mathbb{R}}^{\times }\times {\mathbb{R}}^d$ to be the continuous wavelet transform of a complex Radon measure on ${\mathbb{R}}^d$ under suitable regularity conditions on the wavelets. We define a new norm on the collection of such functions and show that it is isometrically isomorphic to the Banach algebra of complex Radon measures under convolution. We show a wavelet inversion formula for measures, in a restricted sense. Under additional hypothesis, we provide a reconstruction formula for the measure of closed rectangles in ${\mathbb{R}}^d$ using wavelet transforms. We have included supporting examples to substantiate these assumptions.