Convolution product notation

Equation (41) can be viewed as the reciprocal of Eq. (37). It can be rewritten, using convolution product notations, as... 

Then, by injecting this scaling factor into expression 11.36 of the basic quantity, one obtains the general solution under the form of a convolution equation, also called convolution product and notated as a multiplication by means of a star operator (dyadic operator) ... 

The INPAR method for the inversion of moment tensor adopts a point-source approximation. The retrieval of the six components of the moment tensor by waveform inversion is a nonlinear problem anyway linearity can be preserved in the first step of the inversion by considering different time evolutions for each of the six components of the moment tensor, namely, the moment tensor rate functions (MTRFs, Panza and Sarah 2000). The kth component of displacement at the surface is the convolution product of the MTRFs and (medium) Green s function spatial derivatives (hereafter Green s functions) and, using Einstein summation notation, can be written as... 

This is nothing but the electron-vibration interaction in the chosen notation. The quantity h is the three index supervector acting on the vector of nuclear shifts they form the scalar product (.... ..) giving a 10 x 10 matrix, next forming a Liouville scalar product with matrix V. On the other hand, acting on the variations V of the density matrix by forming the Liouville scalar product h produces a vector to be convoluted with that of nuclear shifts 5q. With use of this set of variables the energy in the vicinity of the symmetric equilibrium point becomes ... 

This last equation provides yet another means of representing and manipulating waves. Because of the compactness of its notation, it is frequently used for expressing diffraction relationships. It is also employed when waves must be multiplied, since the product of two exponentials is obtained simply by adding their exponents. This is important in terms of defining convolutions, which are the products of wave functions. We will encounter those in Chapters 5 and 9. 

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