Abstract
Full Text
MATHEMATICS
V. A. DITKIN
OPERATIONAL CALCULI FOR FUNCTIONS DEFINED ON THE WHOLE LINE
(Presented by Academician A. A. Dorodnitsyn, 9 VIII 1956)
We shall say that a function (f(x)) belongs to the set (S) if:
1) (f(x)) is defined almost everywhere on the line (-\infty < x < \infty) and is Lebesgue integrable on every finite interval.
2) There exists at least one pair of values (p) for which the Laplace integrals
[
\int_a^\infty f(x)e^{-p_1x}\,dx
\quad \text{and} \quad
\int_{-\infty}^b f(x)e^{-p_2x}\,dx
]
converge. From the convergence of the integrals there follows the existence of numbers (\sigma_1) and (\sigma_2) such that everywhere in the half-plane (\operatorname{Re} p > \sigma_1) the first integral converges and represents an analytic function, regular in this half-plane. Similarly, the second integral converges everywhere in the half-plane (\operatorname{Re} p < \sigma_2) and in this half-plane represents an analytic regular function.
Denote by (S_+) the set of all functions from (S) that are equal to zero for all (xb). The numbers (a) and (b) depend on the choice of the function (f(x)). The Laplace transform
[
\overline{f}1(p)=\int\,dx,\qquad f_1(x)\in S_+,}^{+\infty} f_1(x)e^{-px
\tag{1}
]
maps the set (S_+) into a set of functions of the complex variable (p=\sigma+i\tau), regular in a right half-plane; we shall denote this set by (\overline{S}_+).
The transform
[
\overline{f}2(p)=-\int\,dx,\qquad f_2(x)\in S_-,}^{+\infty} f_2(x)e^{-px
]
maps the set (S_-) into a set of functions of the complex variable (p=\sigma+i\tau), regular in a left half-plane; we shall denote this set by (\overline{S}_-).
The sets (\overline{S}+) and (\overline{S}-), with the usual definition of addition and multiplication by a complex number, form linear sets. Denote by (\mathfrak{M}) the direct sum of the sets (\overline{S}+) and (\overline{S}-), i.e. the set of all pairs ((\overline{f}1(p),\overline{f}_2(p))), (\overline{f}_1(p)\in \overline{S}+) and (\overline{f}2(p)\in \overline{S}-). In this case the sum of two pairs ((\overline{f}_1(p),\overline{f}_2(p))), ((\overline{g}_1(p),\overline{g}_2(p))) is called the pair ((\overline{f}_1(p)+\overline{g}_1(p),\overline{f}_2(p)+\overline{g}_2(p))), and the product of ((\overline{f}_1(p),\overline{f}_2(p))) by a complex number (\lambda) is called the pair ((\lambda \overline{f}_1(p),\lambda \overline{f}_2(p))).
Consider in (\mathfrak M) the linear subset (\mathfrak M_0), consisting of all pairs of the form ((\bar\theta(p), \bar\theta(p))), where
[
\bar\theta(p)=\int_a^b \theta(x)e^{-px}\,dx;
]
(\theta(x)) belongs to (S_+S_-)—the intersection of the sets (S_+) and (S_-).
Denote by (\bar S) the quotient set (\mathfrak M/\mathfrak M_0). The elements of (\bar S) are classes. Two elements of (\mathfrak M) belong to one and the same class if their difference belongs to (\mathfrak M_0). As is known, (\bar S) will be a linear set.
Theorem. The linear sets (S) and (\bar S) are isomorphic.
Let (f(x)\in S),
[
\bar f_+(p)=\int_0^\infty f(x)e^{-px}\,dx,\qquad
\bar f_-(p)=-\int_{-\infty}^0 f(x)e^{-px}\,dx .
]
To the function (f(x)) we assign the element of the set (\bar S) whose representative is the pair ((\bar f_+(p),\bar f_-(p))), i.e. the coset to which ((\bar f_+(p),\bar f_-(p))) belongs; denote this class by (\bar f). The mapping of (S) into (\bar S) thus constructed will be the desired isomorphism. First of all, it is clear that this mapping is linear; we now prove that every element (\bar g\in\bar S) is the image of some element (g(x)\in S).
Indeed, let ((\bar g_1(p),\bar g_2(p))) be a representative of the class (\bar g). By the definition of the pair ((\bar g_1(p),\bar g_2(p))), there exist functions (g_1(x)\in S_+) and (g_2(x)\in S_-) such that
[
\bar g_1(p)=\int_a^\infty g_1(x)e^{-px}\,dx,\qquad
\bar g_2(p)=-\int_{-\infty}^b g_2(x)e^{-px}\,dx .
]
Recall that (g_1(x)=0) for (xb). The function (g(x)=g_1(x)+g_2(x)) is the preimage of (\bar g\in\bar S). Indeed, if (a<0) and (b<0), then
[
\bar g_+(p)=\int_0^\infty (g_1(x)+g_2(x))e^{-px}\,dx=
]
[
=\int_0^\infty g_1(x)e^{-px}\,dx
=\bar g_1(p)-\int_a^0 g_1(x)e^{-px}\,dx,
]
[
\bar g_-(p)=-\int_{-\infty}^0 (g_1(x)+g_2(x))e^{-px}\,dx=
]
[
=-\int_{-\infty}^b g_2(x)e^{-px}\,dx-\int_a^0 g_1(x)e^{-px}\,dx,
]
[
\bar g_-(p)=\bar g_2(p)-\int_a^0 g_1(x)e^{-px}\,dx.
]
It follows from this that the pairs ((\bar g_+(p),\bar g_-(p))) and ((\bar g_1(p),\bar g_2(p))) belong to the same class (\bar g). Similarly one can show that in the other cases ((a>0,\ b<0;\ a>0,\ b>0;\ a<0,\ b>0)) the pairs ((\bar g_+(p),\bar g_-(p))) and ((\bar g_1(p),\bar g_2(p))) belong to one and the same class.
Thus, the mapping under consideration is a mapping of (S) onto (\overline S). If the image (h(x)) of the element (\overline h) is equal to zero almost everywhere, then the class (\overline h) coincides with the set (\mathfrak M_0). Conversely, if (\overline h) coincides with (\mathfrak M_0), then the image of the element (\overline h) is equal to zero almost everywhere. Indeed, as a representative of (h) one may take the pair ((0,0)); consequently,
[
\int_a^\infty h_1(x)e^{-px}\,dx=0
\quad\text{and}\quad
\int_{-\infty}^b h_2(x)e^{-px}\,dx=0,
]
whence almost everywhere (h_1(x)\equiv 0) and (h_2(x)\equiv 0), and therefore (h(x)\equiv 0). Thus the one-to-one correspondence of the set (S) onto (\overline S) has been proved.
Let us consider pairs ((F_1(p),F_2(p))), where the function (F_1(p)) is equal to the quotient of two functions of the set (\overline S_+), and (F_2(p)) is equal to the quotient of two functions of the set (\overline S_-). We shall denote the set of all such pairs by (\overline{\mathfrak M}).
If the pair ((F_1(p),F_2(p))) belongs to (\overline{\mathfrak M}), then there exist in (\mathfrak M) elements ((\overline f_1(p),\overline f_2(p))) such that the pairs ((F_1(p)\overline f_1(p),F_2(p)\overline f_2(p))) again belong to (\mathfrak M). To each pair ((F_1(p),F_2(p))) of the set (\overline{\mathfrak M}) we shall put in correspondence an operator (F). Denote by (\overline\Omega_F) the totality of all elements ((\overline f_1(p),\overline f_2(p))) of the set (\mathfrak M) for which ((F_1(p)\overline f_1(p),F_2(p)\overline f_2(p))) again belongs to (\mathfrak M). To the set (\overline\Omega_F) there corresponds a certain subset in the set (\overline S). We shall denote the image of this subset under the isomorphism (S\leftrightarrow \overline S) by (\Omega_F). Consequently, if (f(x)) belongs to (\Omega_F), then among the elements of the class (\overline f) there exist such pairs ((\overline f_1(p),\overline f_2(p))) that ((F_1(p)\overline f_1(p),F_2(p)\overline f_2(p))) belongs to (\mathfrak M).
On the set (\Omega_F) we define the operator (F), putting, for (f\in\Omega_F),
[
Ff=g;
]
here (g) is the image of the class (\overline g), whose representative is the pair ((F_1(p)\overline f_1(p),F_2(p)\overline f_2(p))). It is clear that the operator (F) thus defined will be linear, but, generally speaking, not single-valued, i.e. to one and the same function (f(x)) there may correspond an infinite set of functions (Ff). This is explained by the fact that the pairs ((F_1(p)\overline\theta(p),F_2(p)\overline\theta(p))) need not again belong to the set (\mathfrak M_0).
By the set of indeterminacy of the operator (F) we shall mean the image of the set of all pairs ((F_1(p)\overline\theta(p),F_2(p)\overline\theta(p))) under the homomorphism of (\mathfrak M) onto (S), where ((\theta(p),\theta(p))) is any pair belonging to (\mathfrak M_0\overline\Omega_F).
Obviously, the set of indeterminacy of the operator consists of the values of the operator (F) over the zero element of the set (S). The operator (F) corresponding to the pair ((F_1(p),F_2(p))) will be single-valued only in the case when its set of indeterminacy is empty. For this it is necessary and sufficient that the pairs ((F_1(p)\overline\theta(p),F_2(p)\overline\theta(p))) belong to the set (\mathfrak M_0) for all ((\overline\theta(p),\overline\theta(p))) belonging to (\mathfrak M_0\overline\Omega_F).
We shall call the sum of the operators (F) and (G) the operator corresponding to the pair ((F_1(p)G_1(p),F_2(p)G_2(p))), and the product of the operators we shall call the operator corresponding to the pair ((F_1(p)G_1(p),F_2(p)G_2(p))). We denote the sum of the operators by (F+G), and the product by (FG).
In conclusion we give several examples. In those cases when (F_1(p)) and (F_2(p)) can be obtained as analytic continuations of the function (F(p)), we shall write (F(p)f) instead of (Ff).
- Operator (\dfrac{1}{p^n}). In this case
[
\frac{1}{p^n}f
=
\frac{1}{(n-1)!}\int_0^x (x-\xi)^{n-1}f(\xi)\,d\xi
+
\sum_{k=0}^{n-1} a_k x^k.
]
The indeterminacy set of the operator (\dfrac{1}{p^n}) consists of all polynomials of degree not higher than (n-1).
-
The operator (p^n). Its domain of definition consists of functions differentiable (n) times, whose derivative of order (n) belongs to (S), and
[
p^n f(x)=f^{(n)}(x).
] -
Regular operators. Let the function (F(p)) be regular at the infinitely distant point of the complex (p)-plane. Then the operator corresponding to the pair ((F(p), F(p))) is called regular. Such an operator is defined on the whole set (S). The value of the operator can be computed by the formula
[
F(p)f(x)=\sum_{k=0}^{\infty}\frac{a_k}{p^k}f(x)+\frac{1}{2\pi i}\int_{|p|=\rho_1>\rho_0} F(p)\,\bar{\theta}(p)e^{pt}\,dp.
]
Here (\theta(p)) is an arbitrary function from (\mathfrak{M}0) and
[
\sum=F(p),\qquad |p|>\rho_0.}^{\infty}\frac{a_k}{p^k
]
For example, the operator (\dfrac{1}{\sqrt{p^2+\lambda^2}}) is regular; the value of this operator on (f(x)\equiv 1) is equal to
[
\frac{1}{\sqrt{p^2+\lambda^2}}\,1
=
J_0(\lambda x)+\frac{1}{\pi}\int_{-\lambda}^{\lambda}
\frac{\bar{\theta}(i\xi)e^{i\xi x}\,d\xi}{\sqrt{\lambda^2-\xi^2}},
]
where (\bar{\theta}(p)) is an arbitrary function from (\mathfrak{M}_0).
Computing Center
Academy of Sciences of the USSR
Received
8 VIII 1956