MATHEMATICS

M. K. FAGE

INTEGRAL REPRESENTATIONS OF OPERATOR-ANALYTIC FUNCTIONS OF ONE INDEPENDENT VARIABLE

(Presented by Academician V. I. Smirnov, 18 III 1957)

Let

\[ L=\frac{d^n}{dx^n}+p_{n-1}(x)\frac{d^{\,n-1}}{dx^{n-1}}+\cdots+p_0(x) \tag{1} \]

be an ordinary linear differential operator with continuous complex (in particular, real) coefficients, given on an interval \((a,b)\) \((-\infty \leq a < b \leq +\infty)\) of the real number line;

\[ M=\frac{d^n}{dw^n}+q_{n-1}(w)\frac{d^{\,n-1}}{dw^{n-1}}+\cdots+q_0(w) \tag{2} \]

an operator with coefficients analytic in a domain \(G\) of the complex \(w\)-plane.

In the note \((^1)\) a topological ring \(A_{L,x_0}\) of functions was constructed, each of which is operator-analytic with respect to \(z\) in a (its own) neighborhood of a certain (arbitrary but fixed) point \(x_0\in(a,b)\), and it was also indicated that the corresponding ring \(A_{M,w_0}\) for the operator \(M\) is, as a set, simply the set \(A_w\) of functions analytic in a neighborhood of the point \(w_0\). On this basis the equivalence of all ordinary operators of equal order (both of the form (1) and of the form (2)) was proved: a transformation of one operator \(L\) into another was constructed, i.e., such a transformation \(T\) which: 1) maps one ring \(A_{L,x_0}\) isomorphically onto another; 2) carries \(L\) into the corresponding other operator.

In the present note an integral form is constructed for the transformation \(T=T_{M,w_0;L,x_0}\) (in this sense) of the operator \(M\) into the operator \(L\)—according to the scheme originating in the works of J. Delsarte \((^2)\) and A. Ya. Povzner \((^3)\), devoted to second-order operators. Thus every function \(g(x)\in A_{L,x_0}\), i.e. \(L\)-analytic in a neighborhood of \(x_0\), receives an integral representation \(g(x)=Tf(w)\) through a function \(f(w)\), analytic in a neighborhood of \(w_0\). This scheme is based on the theory of the Cauchy problem for the partial differential equation

\[ MF(w,x)=LF(w,x), \tag{3} \]

which must be solved with the initial values

\[ F(w,x_0)=f_0(w),\ldots,\left.\frac{\partial^{\,n-1}F(w,x)}{\partial x^{n-1}}\right|_{x=x_0}=f_{n-1}(w), \tag{4} \]

analytic in the domain \(G\).

§ 1. Local integral representation. Using the results of the note \((^1)\), one can obtain the following theorem:

Theorem 1. For each \(w_0\in G\), the solution \(F(w,x)\) of problem (3), (4) exists in the complex-real cylinder \(C(|w-w_0|<\alpha,\ |x-x_0|<\beta)\) and can be represented in the form of the sum of the double series

\[ F(w,x)=\sum_{\mu=0}^{\infty}\sum_{m=0}^{\infty} a_{\mu,m}g_\mu(w,w_0)f_m(x,x_0), \tag{5} \]

where \(\{g_\mu(w,w_0)\}_0^\infty\) is an \(M\)-basis at the point \(w_0\); \(\{f_m(x,x_0)\}_0^\infty\) is an \(L\)-basis at the point \(x_0\), and the coefficients satisfy the inequalities

\[ |a_{\mu,m}|\leq C_1^\mu C_2^m\mu!m! \tag{6} \]

for all \(\mu,m=0,1,2,\ldots\).*
The series (5) converges absolutely and uniformly inside \(C\) and permits termwise application of the operators \((\partial^\rho/\partial w^\rho)M^\chi\), \((\partial^r/\partial x^r)L^q\) \((\rho,r=0,1,\ldots,n-1;\ \chi,q=0,1,2,\ldots)\) in any order. Hence, from (6) one may obtain the following result:

Theorem 2. The function \(F(w,x)\) of the preceding theorem and its \((\partial^\rho/\partial w^\rho)M^\chi\)-images \((\rho\leq n-1)\), for fixed \(w\), are \(L\)-analytic with respect to \(x\) (in the neighborhood \(|x-x_0|<\beta\)).

Fix \(w_0\in G\), consider some function \(f(w)\in A_w\), and find the solution \(F(w,x)=F_k(w,x)\) of problem (3), (4) for \(f_k(w)=f(w)\), \(f_s(w)=0\) \((s=0,1,\ldots,n-1;\ s\ne k)\). The functions

\[ c_{\mu,k}(x)\equiv c_{\chi n+\rho,k}(x)= \left.\frac{\partial^\rho}{\partial w^\rho}M^\chi F_k(w,x)\right|_{w=w_0} \]

\((\mu=\chi n+\rho)\) will then be \(L\)-analytic in a neighborhood of \(x_0\), i.e., will belong to the ring \(A_{L,x_0}\). This defines the operators \(T_{\mu,k}f(w)=c_{\mu,k}(x)\) \((\mu=0,1,2,\ldots;\ k=0,1,\ldots,n-1)\), transforming \(A_{M,w_0}\) into \(A_{L,x_0}\) and constituting a matrix \(\widetilde T\) with infinitely many rows and \(n\) columns, the elements of which, under the action of the operator \(M\) on the right or \(L\) on the left, are shifted downward by \(n\) rows; that is, in general:

\[ T_{\rho,k}M^\chi=L^\chi T_{\rho,k}=T_{\chi n+\rho,k}\,{}^{**}. \tag{7} \]

By virtue of (7), the matrix \(\widetilde T\) naturally decomposes into square cells \(\widetilde T_0,\widetilde T_1,\ldots\) (from top to bottom) of \(n\times n\) operators in each cell.

Comparing the properties of the operators \(T_{\rho,k}\) \((\rho,k=0,1,\ldots,n-1)\) of the upper cell \(\widetilde T_0\) with the properties of the operator \(T=T_{M,w_0;\,L,x_0}\), constructed in (1), we arrive at the following basic result:

Theorem 3. The trace \(T_{0,0}+\cdots+T_{n-1,n-1}\) of the cell \(\widetilde T_0\) coincides with the transformation \(T\), which isomorphically maps the ring \(A_{M,w_0}\) onto the ring \(A_{L,x_0}\) and carries the operator \(M\) into the operator \(L\).

Thus, a local “integral” representation \(g(x)=Tf(w)\) has been obtained for \(L\)-analytic functions \(g(x)\), which is integral only in the sense that it is composed of solutions (“integrals”) of the Cauchy problem (3), (4).

§ 2. The domain of dependence of the integral representation. Applying the method of increasing the number of independent variables \((^5)\) to the solution of problem (3), (4), in particular, in constructing each function

* With the possible exception of \(\mu=m=0\); here \(C_1\) and \(C_2\) are some constants; one may take \(C_1=C_2\).

** In these relations, apparently, the foundations are laid for extending to the operators (1) the theory of the generalized shift operators of Delsarte \((^2)\)—Levitan \((^4)\).

\(F_k(w,x)\), we obtain an \(L\)-analytic continuation of these functions (in \(x\)) and thereby an \(L\)-analytic continuation of the representation \(g(x)=Tf(w)\):

Theorem 4. If \(f(w)\) is regular in the disk \(|w-w_0|<R\), then the function \(g(x)=Tf(w)\) admits an \(L\)-analytic continuation* to the interval
\((x_0-R,\ x_0+R)\cap(a,b)\).

In the case of sufficiently smooth coefficients of the operator \(L\), applying to each function \(F_k(w,x)\) the generalized Riemann formula (see \((^5)\), § 3), we obtain the following integral (in the ordinary sense) form of the representation \(g(x)=Tf(w)\):

Theorem 5. If each coefficient \(p_k(x)\) of the operator \(L\) is continuously differentiable \(k\) times \((k=0,1,\ldots,n-1)\), then

\[ g(x)=Tf(w)=\sum_I \int_{\Omega_I}\cdots\int K_I(w_0,x;t_I)f(w_{0,I})\,dt_{I_\alpha}. \tag{8} \]

Here, for brevity, by \(I\) is denoted an arbitrary nonempty combination (subset) of numbers \(i_0<i_1<\cdots<i_m\) from the set \(1,2,\ldots,n\); \(\sum_I\) extends over all these combinations; \(K_I(w,x;t)\) is a linear combination of derivatives

\[ \frac{\partial^k}{\partial w^k}\, \frac{\partial^s}{\partial t_{\sigma_1}\cdots\partial t_{\sigma_s}}\, v(w,x;t) \]

of the Riemann function \(v(w,x;t)=v(w,x;t_1,\ldots,t_n)\), where the set of indices
\(\sigma_1<\sigma_2<\cdots<\sigma_s\) runs through \(I\) and \(k+s\le m\); the introduction of \(t_I\) instead of \(t\) as an argument means that the variables \(t_i\) with numbers \(i\notin I\) are set equal to zero; the coefficients of these linear combinations \(K_I(\cdots)\) are regular functions of \(w\); \(w_{0,I}=w_0\pm\sum_{i\in I}t_i\varepsilon_i\)**, where \(\varepsilon_1=1,\ldots,\varepsilon_n\) are all roots of the \(n\)-th degree of 1; finally, the domain of integration \(\Omega_I\) is the \(m\)-dimensional face \((t_i=0\) for \(i\notin I)\), corresponding to \(I\), of the simplex
\((t_1\ge0,\ldots,t_n\ge0,\ t_1+\cdots+t_n=|x-x_0|)\) in the \(n\)-dimensional space of characteristic variables \(t_1,\ldots,t_n\); in integration one of the variables, for example \(t_{i_\alpha}\), is excluded, and then \(dt_{I_\alpha}\) is an abbreviated notation for the product of all differentials \(dt_{i_0},\ldots,dt_{i_m}\), except \(dt_{i_\alpha}\).

The nonintegral term of formula (8), corresponding to \(m=0\)***, is equal to

\[ \exp\left\{\frac{1}{n}\int_{w_0}^{w_0+x-x_0} q_{n-1}(w)\,dw +\frac{1}{n}\int_x^{x_0} p_{n-1}(t)\,dt\right\} f(w_0+x-x_0). \tag{9} \]

For the simplest operator \(M=d^n/dw^n\), the coefficients of the linear combinations \(K_I(\cdots)\) can be computed explicitly in the form of functions of the roots \(\varepsilon_1,\ldots,\varepsilon_n\) and of the values \(p_k^{(\nu)}(x_0)\) \((\nu=0,1,\ldots,k;\ k=1,2,\ldots,n-1)\). The resulting general formula, which we do not write out for lack of space, for \(n=2,\ w_0=0,\ x_0=0,\ L=d^2/dx^2+q(x)\), after a change of variable of integration, coincides with formula (4) in the note by A. Sh. Blokh \((^6)\). For \(n=3,\ w_0=0,\ x_0=0,\ L=d^3/dx^3+p(x)d/dx+q(x)\), we obtain

\[ \text{* Unique by Theorem 6 }(^1). \]

\[ \text{** Plus for } x\ge x_0,\ \text{minus for } x<x_0. \]

\[ \text{*** It would seem that there should be } n \text{ such terms, corresponding to the values } i_0=1,2,\ldots,n;\ \text{but they are all equal to zero, except one, corresponding to } i_0=1 \text{ and equal to (9); this is connected with the fact that } \varepsilon_1=1. \]

\[ \begin{aligned} g(x)=f(x)&+\int_0^{|x|} K_2(x;t)\, f\bigl(\pm[\varepsilon_1 t+\varepsilon_2(|x|-t)]\bigr)\,dt\\ &+\int_0^{|x|} K_3(x;t)\, f\bigl(\pm[\varepsilon_1 t+\varepsilon_3(|x|-t)]\bigr)\,dt\\ &+\int_0^{|x|} ds \int_0^{|x|-s} K_{23}(x;s,t)\, f\bigl(\pm[\varepsilon_1 s+\varepsilon_2 t+\varepsilon_3(|x|-s-t)]\bigr)\,dt . \end{aligned} \tag{10} \]

where \(\varepsilon_1=1,\ \varepsilon_2=-\frac12+i\frac{\sqrt3}{2},\ \varepsilon_3=\bar{\varepsilon}_2;\)
\(K_2(x;t)=\partial v(x;t_1,t_2,0)/\partial t_2\) for \(t_1=t,\ t_2=|x|-t;\)
\(K_3(x;t)=\partial v(x;t_1,0,t_3)/\partial t_3\) for \(t_1=t,\ t_3=|x|-t;\)
\(K_{23}(x;s,t)=\partial^2 v(x;t_1,t_2,t_3)/\partial t_2\partial t_3+p(0)v\) for
\(t_1=s,\ t_2=t,\ t_3=|x|-s-t\). Here the Riemann function
\(v=v(x;t_1,t_2,t_3)\) does not depend on \(w\), and, by virtue of its symmetry in
\(t_1,t_2,t_3\), the functions \(K_2(x;t)\) and \(K_3(x;t)\) are equal to each other.
Thus, in the case \(n=3\) one may say that the integrals in the integral representation
of \(L\)-analytic functions are taken over the triangle with vertices
\(x\varepsilon_1=x,\ x\varepsilon_2,\ x\varepsilon_3\) in the complex \(w\)-plane, along
two of its sides with common vertex \(x\), and “over the vertex” \(x\).

Chernivtsi
State University

Received
15 III 1957

REFERENCES

\({}^{1}\) M. K. Fage, DAN, 112, No. 6 (1956).
\({}^{2}\) J. Delsarte, Acta Math., 69, 259 (1938).
\({}^{3}\) A. Ya. Povzner, Matem. sborn., 23 (65), No. 1 (1948).
\({}^{4}\) B. M. Levitan, Uspekhi matem. nauk, 4, issue 1 (29) (1949).
\({}^{5}\) M. K. Fage, DAN, 108, No. 6 (1955).
\({}^{6}\) A. Sh. Blokh, DAN, 92, No. 2 (1953).