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9. Convolution
 
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The generalizations of the derivative as expressed in (4.6) and (7.4) suggest that they can be formulated in terms of the convolution, which would be important for the convolution is a very simple operation in the frequency spaces achieved by Laplace and Fourier transforms. The following development shows how this is the case, and how after all the derivative of a function is its convolution with certain function,

(9.1)

which Laplace convolution with f(x) yields the Riemann-Liouville derivative of order -a

(9.2)

And since the transform of the function expressed in (9.1) gives the following simple result

(9.3)

and the transform of the convolution is the multiplication of the transforms, in the case that the function f fulfills all the requirements given previously for the simplicity of the Laplace transform of the derivative, we again get the equation (8.4), which was the key for the generalization by Laplace transform

(9.4)

These last results showing that the generalized derivative is a convolution with certain function, open the interesting question of what other kind of operators would be defined if functions other than (9.1) had been chosen. The answer is that the new operators defined would be functions of the derivative. To see this, we can exploit the linearity of the convolution, supposing that the function g can be expanded in powers of x

(9.5)

where

(9.6)

This proves the equivalence between functions of the derivative and convolutions. For any convolution there is an equivalent function of the derivative if the function implied can be expanded in powers of x and vice versa. Now, the Laplace transform shows what was expectable,

(9.7)

since the linearity of the Laplace transform combined with the result of (8.5) shows clearly that

(9.8)

as well as (8.11) shows for the Fourier transform that

(9.9)

These are useful tools for the calculation of functions of the derivative. As an example, the following case is considered with the help of Fourier transforms

(9.10)

and directly as in (5.2)

(9.11)

yielding both methods the same result. The case of (5.3) also matches when calculated with Fourier transforms.