ditionally satisfies for some non-negative integer-valued vector m = ( m$_{1}$, ..., m$_{d}$ ) with ((1 + t $^{2}$) − $^{1}$) m denoting the corresponding product Π d i $_{=1}$((1 + t 2 $_{i}$) − $^{1}$) $^{m$_{i}$}$similarly by (12) de- fines a generalized function b in S $^{∗}$; such generalized functions are called regular functions in the space of generalized functions. ∫ ( (1 + t $^{2}$) − $^{1}$) $^{m}$| b ( t ) | dt < ∞ (13) ( f, ψ ) = − ∫ F ( x ) dψ dx ( x ) dx. (14) Generalized derivatives ∂ $^{m}$b, m = ( m$_{1}$, ...m$_{d}$ ) exist for any generalized function b in D ∗ and S ∗ and are defined as functionals by the value for any function, ψ, as ( ∂ $^{m}$b, ψ ) = ( − 1) $^{m}$( b, ∂ $^{m}$ψ ) (here ( − 1) m = ( − 1) m$_{1}$ + ... + $^{m$_{d}$}$) . The differentiation operator is continuous in S $^{∗}$and in D $^{∗}$. Thus, for example, a generalized density function, f, in the univariate case is defined as a generalized derivative of the regular distribution function, F ( x ) , by providing for any real-valued function, ψ ∈ G, the value Thus defined generalized density is an element of D $^{∗}$, or of S $^{∗}$, moreover, it continuously depends on the distribution functions in the topology of either S ∗ or D ∗ by continuity of the differentiation operator. E.g. if F ( x ) = I ( x ≥ 0) (the indicator function I ( θ ) = 1 if θ is true, zero otherwise), by substituting into (14) we obtain the generalized derivative as the Dirac δ − function that provides ( δ, ψ ) = ψ (0) . Any generalized function can be represented as a generalized derivative 16