The first is the Titchmarsh

Convolution Theorem, which characterizes the null spaces of Volterra convolution operators.

The FRF (3) can be used to derive displacement from a superposition of effects of type (2), by employing the discrete

convolution theorem [13].

Theorem 1 (

convolution theorem: see Theorem 4.1 in [14]).

Lewis, "Applications of a

convolution theorem to Jacobi polynomials," SIAM Journal on Mathematical Analysis, vol.

Li, "

Convolution theorem for fractional cosine-sine transform and its application," Mathematical Methods in the Applied Sciences, vol.

Although description of physical quantities in terms of its Fourier transform enables an approach for inhomogeneous media formally analogous to that of plane waves, its application to constitutive laws require (according to the

convolution theorem: "the Fourier transform of a product of functions is equal to the convolution of their Fourier transforms" [7]) to replace matrix products by cumbersome convolution matrix products in dispersion equation.

(5) we used the Laplace transforms, and by applying the

convolution theorem. [6, 7] we obtain in the expression of the function of renewal:

on applying the

convolution theorem and using ([lambda]/[GAMMA]([alpha]))[[integral].sup.x.sub.0][(x - t).sup.[alpha]-1] [E.sub.[alpha]]([lambda] [t.sup.ta)dt = [E.sub.[alpha]]([lambda][x.sup.[alpha]]) - 1, [alpha] > 0 [12, page 109], we get

It follows from the classical Titchmarsh

convolution theorem and uniqueness theorem for analytic functions that ker ([K.sub.f]) = {0}.

By the inverse of a new integral transform and

convolution theorem (33) we find that:

According to the

convolution theorem, the convolution in one domain can be converted to the point-wise multiplication in another domain.

(82)] based on the generalized

convolution theorem, together with the product formulas derived in [6, Theorem 6] and [7, Theorem 8], it follows that: