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V. A. DITKIN
ON THE THEORY OF OPERATIONAL CALCULUS
(Presented by Academician I. N. Vekua, 27 VI 1958)
In 1950–1951 Mikusiński \((^{1})\) constructed an operational calculus without any connection with the theory of the Laplace transform. Despite all the attractiveness and elegance of this theory, it must be said that in a number of cases, when the Laplace integral is used, various transformations and computations connected with the finding of operational formulas are considerably simplified. Therefore it is expedient to construct, for Mikusiński’s operator calculus, an analogue of the Laplace transform. Such a construction is given in this note. It should be regarded as a generalization of our paper \((^{2})\).
Denote by \(L\) the set of all functions defined on the interval \(0<t<\infty\) and summable in the Lebesgue sense on every finite interval \((0,A)\), \(A>0\). Let \(S\) be the set of all functions in \(L\) for which the Laplace integral
\[ \bar f(p)=\int_{0}^{\infty} f(t)e^{-pt},\qquad p=\sigma+i\tau, \tag{1} \]
converges absolutely, and let \(\bar S\) be the set of their Laplace transforms, i.e., the totality of all functions \(\bar f(p)\), \(f(t)\in S\).
From the known properties of the Laplace integral it follows that \(\bar S\), with respect to the usual operations of addition and multiplication, is a ring. Constants do not belong to the ring \(\bar S\). However, the product \(a\bar f(p)\) belongs to \(\bar S\) for every number \(a\).
Denote by \(\bar J_A\) the totality of all functions in \(\bar S\) representable in the form \(e^{-pA}\bar g(p)\), where \(\bar g(p)\in\bar S\). \(e^{-pA}\bar g(p)\) belongs to \(S\) for every function \(\bar g(p)\in\bar S\). \(\bar J_A\) is an ideal in the ring \(\bar S\). Construct the set \(\bar S_A=\bar S/\bar J_A\). The elements of the set \(\bar S_A\) are the residue classes modulo \(\bar J_A\). As is known, \(\bar S_A\) will also be a ring. We shall denote the elements of the set \(\bar S_A\) by the letters \(\bar F_A,\bar G_A\), etc. Let \(A=1,2,\ldots,n,\ldots\). We obtain a sequence of sets \(\bar S_1,\bar S_2,\ldots,\bar S_n,\ldots\). Form the direct sum of these sets*. As is known, the elements of the direct sum will be systems \(\bar F=\{\bar F_1,\bar F_2,\ldots,\bar F_n,\ldots\}\), where \(\bar F_n\) is an element of the set \(\bar S_n\).
The direct sum is also a ring. If \(\bar G=\{\bar G_1,\bar G_2,\ldots,\bar G_n,\ldots\}\) is another element of the direct sum, then, by definition,
\[ \bar F+\bar G=\{\bar F_1+\bar G_1,\ \bar F_2+\bar G_2,\ldots,\bar F_n+\bar G_n,\ldots\}, \]
\[ \bar F\cdot\bar G=\{\bar F_1\cdot\bar G_1,\ \bar F_2\cdot\bar G_2,\ldots,\bar F_n\cdot\bar G_n,\ldots\},\qquad a\bar F=\{a\bar F_1,\ a\bar F_2,\ldots,\ a\bar F_n,\ldots\}. \]
* Sometimes it is more expedient to consider the direct sum of the rings \(\bar S_A\) over all values of \(A\) from zero to infinity.
In the direct sum we single out the subset consisting of all elements
\[ F=\{\overline F_1,\ \overline F_2,\ldots,\overline F_n,\ldots\}, \]
for which \(\overline F_1 \supset \overline F_2 \supset \cdots \supset \overline F_n \supset \overline F_{n+1}\supset \cdots\). Here the \(\overline F_n\) are regarded as sets from \(\overline S\). We shall denote the collection of all such elements by \(\mathfrak M\). It is not difficult to prove that \(\mathfrak M\) is a ring.
Define in the set \(L\), alongside the usual notion of the sum of functions and the product of a function by a number, the product
\[ f(t)*g(t)=\int_0^t f(t-\xi)g(\xi)\,d\xi,\qquad f\in L,\quad g\in L. \tag{2} \]
As is known, the product \(f(t)*g(t)\) belongs to \(L\), and with this definition of multiplication \(L\) will be a commutative ring.
Let \(f(t)\in L\). Consider the integral
\[ \overline f_n(p)=\int_0^n f(t)e^{-pt}\,dt. \tag{3} \]
Let \(\overline F_n\) be the complex class containing the function \(\overline f_n(p)\). Since
\[ \overline f_{n+1}(p)-\overline f_n(p) = e^{-np}\int_0^1 f(n+\xi)e^{-p\xi}\,d\xi\in \overline J_n, \]
we have \(\overline F_n\supset \overline F_{n+1}\). Therefore
\[ F=\{\overline F_1,\overline F_2,\ldots,\overline F_n,\ldots\}\in\mathfrak M. \]
One can show that the mapping thus established is a one-to-one mapping of \(L\) onto \(\mathfrak M\).
Theorem 1. The rings \(L\) and \(\mathfrak M\) are isomorphic.
The proof of this theorem rests on the identity
\[ \overline h_n(p)=\overline f_n(p)\,\overline g_n(p) - e^{-np}\int_0^n e^{-pt}\,dt\int_t^n f(t-\xi+n)g(\xi)\,d\xi. \]
Here \(\overline g_n(p)\) is defined in the same way as \(\overline f_n(p)\) (see (3)), and
\[ \overline h_n(p)=\int_0^n h(t)e^{-pt}\,dt,\qquad \text{where }\quad h(t)=\int_0^t f(t-\xi)g(\xi)\,d\xi. \]
This identity shows that \(\overline h_n(p)\) is congruent to \(\overline f_n(p)\overline g_n(p)\) modulo the ideal \(\overline J_n\).
The ring \(L\) has no zero divisors. Extending it to its quotient field, we obtain the ring \(R(L)\)—the field of Mikusiński operators. It follows from Theorem 1 that \(\mathfrak M\) has no zero divisors, and the extension \(R(\mathfrak M)\) will be isomorphic to the field of operators \(R(L)\). The quotient field \(R(\mathfrak M)\) is not difficult to construct directly, and the apparatus of the theory of functions of a complex variable can be brought to bear on its study. In particular, by using the properties of the indicator for entire functions of completely regular growth, one can directly prove the absence of zero divisors in the ring \(\mathfrak M\).
Computing Center
Academy of Sciences of the USSR
Received
23 VI 1958
References
- J. Mikusinski, Rachunek operatoròw, Warszawa, 1953.
- V. A. Ditkin, Uspekhi Mat. Nauk, 2, no. 6 (1947).