ABSTRACT
D efin ition 11.1. All additive, homogeneous and continuous functionals (ζ(μ), Я), defined on î>, are called generalized functions (GF). The set of GF is denoted by V ' . The sequence of GF Rk Є V' converges to Я Є V , if lim (ζ(μ)} Я*) = (ζ(μ),ϋΐ), к —► oo for any ζ(μ) Є V. A GF R Є V' is called regular if it is determined by the following formula
D efinition 11.2. Let the function f ( t ) be locally integrable for t > 0 (i.e., it is integrable on every finite segment [Ο,σ]). A GF £ /(μ) Є V ' defined by the equality
is called a generalized Fourier-Laplace transformation for the function f(t). As ζ(μ) Є V in (11.1), so ζ(μ) Є ^ 2(—οο,οο). Therefore, by virtue of the PaleyWiener theorem, the function
is continuous and finite. Consequently, the integral in (11.1) exists. We note that f ( t ) Є £(0,oo) implies
In this case Lf (μ) is, consequently, a regular GF and coincides with the ordinary Fourier-Laplace transformation for the function f(t). Since
the following inversion formula occurs:
11.2. In this item we give the formulation and solution of the inverse problem for differential operators of the third order, in order to simplify calculations. The general case of arbitrary order operators will be briefly described in item 11.3.
