Integral transforms convolution property

Laplace transforms and the properties of the integral of convolution permit us to establish simple relationships between distinct functions defined in the present context such as those outlined in Chapter 5, Thus,... 

Integral transforms can be used to solve ordinary differential equations by converting them to algebraic equations. In what follows, the convolution properties of the different transforms have been listed, followed by the methods of integral transform used to solve (a) one-dimensional diffusion equations in the infinite and semi-infinite domains and (b) Laplace equations in the cylindrical geometries. 

There are good reasons for choosing to carry out certain operations in the Laplace domain rather than performing equivalent operations in the time domain. In particular, integration of a function with respect to time is equivalent in the Laplace domain to division by the Laplace variable s. Conversely, differentiation corresponds to multiplication by s. This latter property enables differential equations to be Laplace transformed and then solved by algebraic means. These Laplace domain operations are all more simple than their time domain counterparts. In addition convolution in the time domain is equivcdent to multiplication in the Laplace domain. Formally, this may be represented by eqn. (24), the left-hand side of which is termed the convolution integral. 

Thus tak [fW[0iT] i which is the convolution integral of the transforms JS[f ] and W[0,T]]. The latter has rather unpleasant properties. For example, Fig. 4.5 shows the even square window and its (purely real) transform. is complex valued, but has... 

It is possible to suggest that relativistic effects are operating within each wave-particle conceived as a four-dimensional space-time continuum, but that the equations of relativity should be inserted within those equations, descriptive of the properties of holographic matrices convolutional integrals and Fourier transformations. 

It is readily shown that several simple mathematical operations such as differentiation, integration, and linear transformations (scaling and translation), as well as more complex operations such as convolution, deconvolution, and Laplace transformations (and inverse Laplace transformation) have the above linear operator property. 

The FT has many useful mathematical properties including linearity, and using eqn [2], the derivative d/( )/d has the transform (i2nv)P v). The convolution theorem is particularly useful because it relates the product of two functions, F(v) G(v), in the frequency domain to the convolution integral... 

---