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The analytical solving dynamic problems of elasticity theory for piecewisehomogeneous half-space is found. The explicit construction of direct andinverse Fourier's vector transform with discontinuous coefficients ispresented. The technique of applying Fourier's vector transform withdiscontinuous coefficients for solving problems of mathematical physics in theheterogeneous environments is developed on an example of the dynamic problemsof the elasticity theory.

Matrix Fourier transform in dynamic theory of elasticity of piecewise homogeneous medium