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

摘要

We identify the result of the continuous wavelet transform with the difference of solutions of two hyperbolic partial differential equations, for which wavelet's shift and scale are considered as independent variables on 2D plane. The characteristic property, which follows from the introduced representation is the fact that the transform's values inside the triangle defined by two characteristics (a = const, b = const) and crossecting the slopped line on a scale-shift plane (a, b) are completely and uniquely defined by the values of the transform and its derivatives along the last mentioned line.