Continuous wavelet transform with the Shannon wavelet from the point of view of hyperbolic partial differential equations (Q343295)

The complex Shannon wavelet is defined as \(\psi(t) = \operatorname{sinc}(t)\, \exp(-2\pi i t)\) for \(t\in {\mathbb R}\). The authors consider the continuous wavelet transform \(W(a,b)\) for the wavelet \(\psi\), where \(a>0\) is the scale and \(b\in {\mathbb R}\) is the shift parameter. Then this wavelet transform is represented in the form \(W(a,b) = W_1(a,b) - W_2(a,b)\), where \(W_1\) and \(W_2\) are solutions of Cauchy problems of hyperbolic partial differential equations.