Justification of the wavelet-based integral representation of a solution of the wave equation

We study the previously obtained integral representation of a solution of the wave equation. The integrand contains the weighted localized solutions of the wave equation, which depend on integration variables. We construct the parameterized family of the localized solutions from the chosen one by employing the transformations of shift, dilation and the Lorentz transform. We give the sufficient conditions of the pointwise convergence for the obtained improper integral. The convergence in the $\mathcal L_2$ sense is proven too.