Justification of a Wavelet-Based Integral Formula for Solutions of the Wave Equation

An integral representation of solutions of the wave equation obtained earlier is studied. The integrand contains weighted localized solutions of the wave equation that depend on parameters, which are variables of integration. Dependent on parameters, a family of localized solutions is constructed fr...

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Bibliographic Details
Published in:Journal of mathematical sciences (New York, N.Y.) N.Y.), 2019-05, Vol.238 (5), p.630-640
Main Authors: Gorodnitskiy, E. A., Perel, M. V.
Format: Article
Language:English
Subjects:
Bibliographies
Convergence
Dependent variables
Integrals
Lorentz transformations
Mathematics
Mathematics and Statistics
Parameters
Wave equations
Wavelet analysis
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Summary:An integral representation of solutions of the wave equation obtained earlier is studied. The integrand contains weighted localized solutions of the wave equation that depend on parameters, which are variables of integration. Dependent on parameters, a family of localized solutions is constructed from one solution by means of transformations of shift, scaling, and the Lorentz transform. Sufficient conditions are derived, which ensure the pointwise convergence of the obtained improper integral in the space of parameters. The convergence of this integral in ℒ 2 norm is proved as well. Bibliography: 22 titles.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-019-04262-5