Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Institution: Cochin University of Science and Technology
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223227691