UDC 517.948.35

ON CERTAIN PROBLEMS IN THE ANALYTIC THEORY OF EQUATIONS OF THE FIRST KIND

Yu. T. Antokhin

The paper studies the problem of solving an equation of the first kind

\[ Ax=f, \tag{1} \]

where \(x, f\) are elements of some Hilbert space, and \(A\) is a linear operator for which zero is a point of the spectrum. This problem belongs to the class of ill-posed problems, whose theory is currently being developed. Various approaches to the theory of equation (1) are contained in the papers [1—4, 11—13, 9]. The usual formulation of the problem is as follows. It is assumed in advance that a solution of equation (1) exists and is unique. It is required to find it. In this case the possibility of solving is connected with additional a priori requirements of various kinds. Thus, for example, in [1] equation (1) is studied for the case when \(A\) is a completely continuous operator and a compact set to which the solution belongs is known in advance. In [11] the possibility of constructing an approximate solution is connected with the fact that, in addition to equation (1), there is a sufficiently good functional, from whose boundedness compactness follows. These and similar requirements, as a rule, are not too restrictive; however, they are external with respect to equation (1).

In the present paper equation (1) is studied under the assumption that the solution exists and is unique, and that \(A\) is an unbounded self-adjoint operator for which zero is a point of the spectrum. Various formulas are obtained for the solution of equation (1) under the assumption that \(f\) belongs to the domain of definition of the operators \(A, A^2, \ldots\), and also under less stringent restrictions. The solutions are given in the form of series in the elements \(A^n f\), whose theory is similar to the theory of expansions in eigenvectors. As in the latter, if one additionally assumes that the desired solution is “sourcewise representable,” i.e. \(x=Ay\), and an estimate for \(y\) is known, then an exact estimate of convergence of the series for \(x\) can be given. There may occur a case in which the operator \(A^{-1}\) exists but is unbounded. Then only the rate of convergence of the series for \(x\) is estimated. It is shown that the construction of various formulas for finding the solution can be based on the ill-posed problem of constructing a function \(F(z)\), analytic in the half-plane \(\operatorname{Re} z>0\), from its known values \(F(n)\), \(n=1,2,\ldots\). The paper obtains conditions for well-posedness of the latter problem and gives the corresponding estimates. The paper establishes a connection between the problem of solving equation (1) and the problem of stabilization for equations of parabolic type. Equations of the second kind, close in a certain sense ...

to equation (1). For example, formulas have been obtained that give a solution of the Fredholm equation \(x-\lambda A x=f\) in the case when \(\lambda\) is not an eigenvalue. The question of an a priori estimate of solutions of equation (1) is considered. For some operators \(A\) an operator \(B\) is constructed such that

\[ (Ax, Bx)>0 \]

for a sufficiently broad class of elements \(x\). From an inequality of the type indicated, a priori estimates are obtained, in particular for the Helmholtz equation in an unbounded domain.

§ 1. SOME FORMULAS SOLVING AN EQUATION OF THE FIRST KIND

1. Let us consider the equation of the first kind (1)

\[ Ax=f, \]

where \(x, f\) are elements of a certain Hilbert space \(H\), and \(A\) is a linear operator for which zero is a point of the spectrum, but is not an eigenvalue, i.e., there exists a sequence \(\{x_n\}\), \(n=1,2,\ldots\), such that \(\|x_n\|=1\), \(\|Ax_n\|\to 0\) as \(n\to\infty\), and \(Ax\ne 0\) for \(x\ne 0\). In what follows it is assumed that a solution of equation (1) exists.

We first present some heuristic methods for obtaining formulas for finding the solution of equation (1), which will then be rigorously proved. Suppose that \(A\) is a positive, unbounded, self-adjoint operator with everywhere dense domain of definition \(D(A)\). Let \(\{E_\lambda\}\) be the resolution of the identity generated by the operator \(A\) (see [10], pp. 230—252). Let \(x\) be a solution of equation (1). Then

\[ x=\int_0^M dE_\lambda x, \tag{2} \]

where \(M\) is the upper bound of the spectrum, \(M=\sup_{\|x\|=1}(Ax,x)\), \(M\leqslant\infty\). To express \(x\) in terms of \(f\), expand the identity (function) in positive powers of \(\lambda\), \(\lambda\geq 0\). For example, let \(M\leqslant 1\). Then one may write the identity

\[ 1=\lambda \sum_{n=0}^{\infty}(1-\lambda)^n,\quad 0<\lambda<1. \tag{3} \]

Using the formula \(A^{n+1}x=A^n f\), from the expansion (3) for a solution of equation (1) we obtain the formula

\[ x=\sum_{n=0}^{\infty}(E-A)^n f, \tag{4} \]

where \(E\) is the identity operator.

Method (4) is analyzed in detail from another point of view in the book [1], p. 20, for the case when \(A\) is a completely continuous operator. The heuristically obtained formula (4) can then be rigorously proved, as will be done below.

Let us give one more formula for solving equation (1), suitable also in the case when \(A\) is an unbounded operator. In order to obtain a formula for the solution in this case, it is necessary to write an expansion of the identity in powers of \(\lambda\), suitable on any interval of the real-

axis. For this purpose one may, for example, take the identity, valid for any \(\lambda>0\),

\[ 1=-\sum_{n=1}^{\infty}\frac{(-1)^n}{n!}\lambda(\lambda-1)\cdots(\lambda-n+1). \tag{5} \]

Expansion (5) leads to a formula for the solution of an equation of the following form:

\[ x=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n!}(A-E)\cdots(A-(n-1)E)f. \tag{6} \]

Solving equation (1) by formula (6) is equivalent to the method of successive approximations according to the scheme

\[ x_1=f,\quad x_n=\frac{1}{n}f+x_{n-1}-\frac{1}{n}Ax_{n-1}. \tag{7} \]

Formulas (3)—(7) may be replaced by others that better take into account the specifics of the operator \(A\). Let us pass to rigorous proofs. We shall prove formula (4) under the assumption that \(A>0\), \(\|A\|\leq 1\), and equation (1) is solvable. From the identity

\[ x=\sum_{n=0}^{N}(E-A)^nAx+(E-A)^{N+1}x, \]

where \(N\) is an arbitrary positive integer, we obtain

\[ \|x-x_N\|^2=\int_{0}^{M}(1-\lambda)^{2N+2}\,d(E_\lambda x,x), \tag{8} \]

where \(x_N=f+\sum_{n=1}^{N}(E-A)^n f\). Further, for any \(\varepsilon\), \(0<\varepsilon<M\), we have

\[ \|x-x_N\|^2=\int_{0}^{\varepsilon}+\int_{\varepsilon}^{M}\leq \int_{0}^{\varepsilon} d(E_\lambda x,x)+(1-\varepsilon)^{2N+2}\|x\|^2. \tag{9} \]

For any \(\delta>0\) and fixed \(x\) one can find such an \(\varepsilon_1\) that

\[ \int_{0}^{\varepsilon_1} d(E_\lambda x,x)<\delta/2. \]

Then, with \(\varepsilon_1\) fixed, we find such an \(N\) that the second term in inequality (9) is less than \(\delta/2\). Then \(\|x-x_N\|^2\leq\delta\), and formula (4) is proved.

We shall prove formula (6) under the assumption that \(A\) is a self-adjoint operator, \(A>0\), equation (1) is solvable, \(f\) belongs to the domain of definition of all operators \(A^n\), \(f\in D(A^n)\), \(n=1,2,\ldots\), and

\[ \|A^n f\|\leq K, \tag{10} \]

where \(K\) does not depend on \(n\), \(n=0,1,2,\ldots\),

\[ \|x\|\leq K. \tag{11} \]

Introduce the notation:

\[ x_N=\sum_{n=1}^{N}\frac{(-1)^{n+1}}{n!}(A-E)\cdots(A-(n-1)E)f, \]

\[ R_N=r_N(A)x,\qquad r_N(\lambda)=\frac{(-1)^N}{N!}(\lambda-1)\cdots(\lambda-N). \]

It is not difficult to verify by induction that if \(x\) is a solution of equation (1), then

\[ x=x_N+R_N . \tag{12} \]

Using the well-known formulas for estimating norms of functions of an operator (see [10], pp. 250–254), we obtain

\[ \|R_N\|^2=\int_0^M |r_N(\lambda)|^2\,d(E_\lambda x,x), \tag{13} \]

where \(M<\infty\). It is not difficult to establish that the following estimates are valid for any \(\varepsilon,\ 0<\varepsilon<1\): \(|r_N|\leqslant 1\) for \(0\leqslant \lambda\leqslant \varepsilon\); \(|r_N|\leqslant K_1(\varepsilon)/N^\varepsilon\) for \(\varepsilon\leqslant \lambda\leqslant N+1-\varepsilon\); \(|r_N|\leqslant \lambda^N/N!\) for \(N<\lambda\). Hence

\[ \|R_N\|^2\leqslant \int_0^\varepsilon d(E_\lambda x,x) +\left(\frac{K_1^2}{N^{2\varepsilon}}+\frac{1}{(N!)^2}\right)K^2 . \]

Now we repeat the argument used in estimating the right-hand side of inequality (9), and prove that \(\|R_N\|\to 0\) as \(N\to\infty\). Formula (6) is proved.

Let us note that formula (6) is also valid in the case when zero is not a point of the spectrum of the operator \(A\). It may be applied, for example, to solving the equation \(\Delta u=f\) in a bounded domain \(G\) with boundary \(\Gamma\), \(u|_\Gamma=0\), \(\|u\|=\|u\|_{L_2(G)}\). Requirement (10) here reduces to the condition that all higher derivatives of the function \(f\) be bounded in the mean; more precisely, that \(\|\Delta^n f\|\leqslant K\) and that \(\Delta^n f=0\) on the boundary \(\Gamma\).

p. 2. The following device for finding formulas for the solution of equation (1) is the most general. Under the assumption that a solution of (1) exists and \(f\in D(A^n)\), \(n=1,2,\ldots\), \(A>0\), one may, from the elements \(A^n x\), \(n=1,2,\ldots\), construct the analytic operator-valued function \(A^z x\) in some connected domain of the \(z\)-plane, knowing the values of this function at the points \(z=1,2,3,\ldots\). The theory of interpolation of entire functions makes it possible to write a constructive formula solving this problem. Letting \(z\) tend to zero in the formula for \(A^z x\), we obtain a formula for the solution of equation (1). It then remains to justify this formula rigorously. Let us give an example.

Let it be required to solve the equation

\[ Ax=f(t), \tag{14} \]

where \(0\leqslant t\leqslant 1\),

\[ Ax=\int_0^1 A(t,s)x(s)\,ds, \]

\(A(t,s)\) is a continuous function for \(0\leqslant s\leqslant 1,\ 0\leqslant t\leqslant 1\), \((Ax,x)>0\) for \(x\ne 0\), \(A(t,s)=A(s,t)\). Assuming that the above scheme indeed gives a solution, let us prove, for example, that

\[ x(t)=-\lim_{\lambda\to-\infty}\lambda x_\lambda(t), \tag{15} \]

where \(x_\lambda(t)\) satisfies the equation of the second kind

\[ x_\lambda-\lambda A x_\lambda=f . \tag{16} \]

We shall also prove that for any fixed \(t_0,\ 0<t_0<1\), the numbers
\(A^n x(t_0)=A^n x(t)|_{t=t_0}\), where
\[ A^n x=\int_0^1 A_n(t,s)x(s)\,ds \]
are iterated kernels, are the values, for \(z=n,\ n=1,2,\ldots\), of a certain function \(F(z)\), analytic in the right half-plane; moreover,
\[ |F(a+iy)|\le A\exp\bigl((\pi-\varepsilon)|y|\bigr), \tag{17} \]
\[ |F(z)|\le B\exp(a|z|) \tag{18} \]
for \(\operatorname{Re} z\ge a_1\), for some positive numbers \(A,B,\varepsilon,a,a_1\) and any fixed \(a\ge a_1\). To this end we construct \(F(z)=F(z;x(t),t_0)\) by the formula (see [5], p. 149, Theorem 2.8.4, where in the right-hand side we put \(\Phi=\sin \pi z/\pi z\) and correct, apparently, an erroneous plus sign to a minus)
\[ F(z)=-\frac{\sin \pi z}{\pi z}\int_{-\infty}^{+\infty} P(\xi)\exp(-z\xi)\,d\xi, \tag{19} \]
where \(P(z)\), for sufficiently large \(\operatorname{Re}z<0\), is given by the expression
\[ P(z)=\sum_{n=1}^{\infty} F(n)(-1)^n n\exp(nz). \tag{20} \]

For the validity of (17), (18), (19), (20), it is necessary and sufficient that the expansion (20) admit analytic continuation to the strip \(|\operatorname{Im}z|<\varepsilon\) and be bounded inside this strip (see [5]). To obtain formulas (15), (16), we put in (20)
\[ F(n)=A^n(x;t)=\int_0^1 A_n(t,s)x(s)\,ds, \tag{21} \]
where \(t\) is fixed, \(t=t_0\), and \(A_n\) are the iterated kernels of equation (16) (see, for example, the book [6]).

Let \(\lambda=-\exp z\). Then from (20) and (21), using the formula
\[ \sum_{m=1}^{\infty}\lambda^{m-1}A_m(t,s)=\Gamma(t,s;\lambda), \]
where \(\Gamma\) is the resolvent of equation (16), we obtain
\[ P(z)=\frac{d}{dz}\left\{\lambda f(t)+\lambda^2\int_0^1\Gamma(t,s;\lambda)f(s)\,ds\right\}. \tag{22} \]
But
\[ f+\lambda\int_0^1\Gamma(t,s;\lambda)f(s)\,ds=x_\lambda(t) \]
is a solution of equation (16). Therefore, for any real \(z\), i.e. negative \(\lambda\), the function \(P(z)\) exists and
\[ P(z)=\frac{d}{dz}\{\lambda x_\lambda(t)\},\qquad \lambda=-\exp z. \tag{23} \]

From our assumptions concerning \(A(t,s)\), with the aid of expansions in eigenfunctions and (23), it is easy to derive that the function \(P(z)\) is analytic and bounded in any strip \(|\operatorname{Im}z|<\pi-\varepsilon,\ 0<\varepsilon\), and \(P(z)\to0\)

as \(z\to\infty\). Therefore formulas (17)—(19) are proved. Assuming that the solution of equation (14) is given by the formula

\[ x(t)=\lim_{z\to 0}F(z)=\int_{-\infty}^{\infty}P(\xi)\,d\xi, \]

from (23) we obtain formula (15).

Let us prove formulas (15), (16) under the assumption that \(A\) is a self-adjoint operator with an everywhere dense domain of definition, \(A>0\). Putting \(\varepsilon=-1/\lambda\), we shall prove for this purpose that if

\[ Ax_0=f, \tag{24} \]

then

\[ x_0=\lim_{\varepsilon\to +0}x_\varepsilon, \tag{25} \]

where

\[ \varepsilon x_\varepsilon+Ax_\varepsilon=f. \tag{26} \]

It is not difficult to prove that if (26) is satisfied, then \(\varepsilon y+Ay=x_0\), where \(x_\varepsilon=x_0-\varepsilon y\), and \(y,x_\varepsilon\) exist and are unique. Therefore, for any \(\delta\), \(0<\delta<M\), we have

\[ \begin{aligned} \|x_\varepsilon-x_0\|^2 &=\varepsilon^2\|y\|^2 =\varepsilon^2\int_0^M \frac{d(E_\lambda x_0,x_0)}{(\lambda+\varepsilon)^2}\le \\[4pt] &\le \int_0^\delta d(E_\lambda x_0,x_0)+\frac{\varepsilon^2}{\delta^2}\|x_0\|^2 . \end{aligned} \tag{27} \]

This implies (25). Formulas (25), (26) were proposed in works [1, 11], etc.

p. 3. Let us make a remark concerning the possible rate of convergence of the expansions (4), (6) and of equality (25). The estimates of the approximations obtained above are not effective. However, if one additionally assumes that the desired solution of equation (1) is “sourcewise representable,” i.e. \(x_0=Ay_0\)*) for some \(y_0\in H\), then the rate of the approximations can be estimated effectively, provided an estimate for \(\|y_0\|\) is known. For example, instead of estimate (27) we would have the estimate

\[ \|x_\varepsilon-x_0\|^2 =\varepsilon^2\|(\varepsilon E+A)^{-1}Ay_0\|^2 \le \varepsilon^2\|y_0\|^2, \]

i.e. \(\|x_\varepsilon-x_0\|\le \varepsilon\|y_0\|\). Instead of inequality (9), for formula (4) we obtain that

\[ \|x-x_N\|^2 =\int_0^M \lambda^2(1-\lambda)^{2N+2}\,d(E_\lambda y_0,y_0) \le \frac{1}{N^2}\|y_0\|^2, \]

i.e. \(\|x-x_N\|\le \dfrac{1}{N}\|y_0\|\). Analogous estimates can also be obtained in the case where \(x=A^\alpha y_0\), \(\alpha>0\). We note that in equation (1) the inverse operator \(A^{-1}\) may exist but be unbounded; for example \(A=-\Delta\), where \(\Delta\) is the Laplace operator defined on functions of the class \(W_2^2(R_n)\); \(R_n\) is \(n\)-dimensional space (see [7], p. 575). The above

*) For unbounded operators this requirement is a substitute for the usual requirement that the solution belong to a compact set (see [1]).

the arguments show that in this case, in formulas (4), (6), and (25), the rate of approximation can be indicated. The derivation of formulas giving the solution of equation (1) can be connected with the problem of stabilization of solutions of equations of parabolic type in Hilbert space. Indeed, if \(x\) satisfies (1), \(A=A^*\), \(A>0\), then consider the equation

\[ u_t=-Au \tag{28} \]

for \(t>0\) and the initial condition \(u(0)=x\). Writing the solution of equation (28) in the form

\[ u(t)=(1+\varepsilon)^A x \]

for \(t=-\ln(1+\varepsilon)\), \(-1<\varepsilon<0\), expanding \(u(t)\) in powers of \(\varepsilon\), assuming that condition (10) is satisfied, and passing to the limit as \(\varepsilon\to -1\), we obtain formula (6), since \(u(t)\to 0\) as \(t\to+\infty\). In the theory of equation (28), estimates from above and below are known for \(\|u(t)\|\) as \(t\to\infty\) (see [17]). In particular, \(u(t)\to0\) as \(t\to\infty\) if \(x=Ay_0\), since in this case

\[ \|u(t)\|^2=\int_0^\infty \lambda^2\exp(-t\lambda)d(E_\lambda y_0,y_0)\leq \frac{1}{(et)^2}\|y_0\|^2. \]

Consequently, the solution of equation (1) is approximately given by the formula

\[ x(\varepsilon)=-\sum_{n=1}^{\infty}\frac{\varepsilon^n}{n!}(A-E)\ldots(A-(n-1)E)f, \]

\(x(\varepsilon)\to x\) as \(\varepsilon\to -1\), and, knowing additional properties of the solution \(x\) and of the operator \(A\), one can write unimprovable two-sided estimates for \(\|x-x(\varepsilon)\|\) as \(\varepsilon\to -1\).

§ 2. EQUATIONS CLOSE TO AN EQUATION OF THE FIRST KIND

1. Let us consider some methods for obtaining formulas giving the solution of the equation \(Ax=f\) in the case when zero is located close to the spectrum of the operator \(A\). Let the solution of the equation

\[ x-\varepsilon Ax=f, \tag{29} \]

be sought, where \(A\) is a self-adjoint operator in the complex Hilbert space \(H\) with an everywhere dense domain of definition \(D(A)\), \(x\in H\), \(f\in H\), \(\varepsilon=\varepsilon_0+i\delta\), \(\delta>0\). The operator \(A\) may be unbounded. Suppose that \(f\in D(A^n)\), and that condition (10) is satisfied. Consider the functions \(\rho_n(\varepsilon)\) and \(\varphi_n(A)\), where

\[ \varphi_n(A)=A(A-iE)\ldots(A-i(n-1)E), \tag{30} \]

\[ \rho_n(\varepsilon)=\frac{\varepsilon^n}{(1-i\varepsilon)\ldots(1-in\varepsilon)}, \tag{31} \]

where \(E\) is the identity operator, \(n=1,2,\ldots\).

Theorem 1.1. If condition (10) is fulfilled, then the solution of equation (29) is given by the formula

\[ x=f+\sum_{n=1}^{\infty}\rho_n(\varepsilon)\varphi_n(A)f. \tag{32} \]

Proof. If \(x\) is a solution of equation (29), then the identity, easily established by induction,

\[ x=f+\sum_{n=1}^{N}\rho_n(\varepsilon)\varphi_n(A)f+r_N, \tag{33} \]

holds, where \(N\) is any integer, \(N\geqslant 1\),

\[ r_N=\varepsilon\rho_N(\varepsilon)(E-\varepsilon A)^{-1}\varphi_{N+1}(A)f. \tag{34} \]

Let us estimate \(\|r_N\|\). First note that

\[ \left| \frac{z(z-1)\ldots(z-N)} {\zeta(\zeta-1)\ldots(\zeta-N)} \right| \leqslant \frac{K_1(a)(1+|z|^{N+1})}{aN^\alpha}, \tag{35} \]

where \(iz\) is real, \(\operatorname{Re}\zeta=-a<0\), \(K_1\) depends only on \(a\), and \(K_1\to K_1(0)\) as \(a\to 0\). Estimates of the type (35) are presented in the book [8], and we shall not dwell on them. Putting in inequality (35) \(z=-i\lambda\), \(\zeta=-\dfrac{i}{\varepsilon}\), we obtain

\[ \left|\varepsilon\rho_N(\varepsilon)\varphi_{N+1}(\lambda)\right| \leqslant \frac{K_1(1+|\lambda|^{N+1})}{aN^\alpha}, \tag{36} \]

where \(a=\delta/(\varepsilon_0^2+\delta^2)\). Using the estimate \(|1-\varepsilon\lambda|\geq \alpha|\varepsilon|\), estimate (36), and condition (10), we finally obtain

\[ \|r_N\|\leqslant \frac{K_2(a)}{a^3N^\alpha}. \tag{37} \]

It follows from this estimate that \(\|r_N\|\to 0\) as \(N\to\infty\) for fixed \(a>0\). The theorem is proved.

In the case when the operator \(A\) is completely continuous and \(\|A\|\leqslant 1\), formula (32) and estimate (37) for the remainder of expansion (32) make it possible to obtain approximate formulas for the solution of equation (29) in the case when the parameter \(\varepsilon\) is large, but does not lie on the spectrum.

Remarks.

1. The method of proof of Theorem 1.1 is applicable to the construction of a solution of an equation of the form \(Bx-\varepsilon Cx=f\), if there exists an operator \(B^{-1}\), \(B=B^*\), \(C=C^*\), \((Bx,x)>0\) for \(x\ne 0\). This equation can be written in the form \(x-\varepsilon B^{-1}Cx=B^{-1}f\), one can use the self-adjointness of the operator \(A=B^{-1}C\) in the scalar product \((x,y)_1=(Bx,y)\), and apply Theorem 1.1.

2. If in Theorem 1.1 it is additionally known that \(A\) is a positive operator, then equation (29) can also be solved for real, but negative, \(\varepsilon\). Namely, if the conditions (10) are satisfied, then the solution of the equation

\[ x+\varepsilon Ax=f \]

for \(\varepsilon>0\), \(A=A^*>0\), is given by the formula

\[ x=\sum_{n=0}^{N}\psi_n(\varepsilon)\omega_n(A)f+r_N, \]

where \(N\) is arbitrary, \(\|r_N\|\to 0\) as \(N\to\infty\),

\[ \psi_n(\varepsilon)=\frac{(-1)^n\varepsilon^n}{(1+\varepsilon)\ldots(1+n\varepsilon)}, \]

\[ \omega_n(A)=A(A-E)\ldots(A-(n-1)E)f, \]

\[ r_N=\varepsilon|\psi_N|(E+\varepsilon A)^{-1}A(A-E)\ldots(A-NE)f, \]

\[ \|r_N\|\leq \varepsilon K\left(\frac{\varepsilon}{1+\varepsilon}\right)^N. \]

1. Let us consider the question of the asymptotics of the solution of equation (29) as \(\varepsilon\to 0\). Let \(f\in D(A^n)\), \(n=1,2,\ldots\), and suppose that for equation (29) the following condition is satisfied (the principle of limiting absorption): for every \(\varepsilon_0\) there exists an element \(x_{\varepsilon_0}\in D(A)\) such that \(\|x-x_{\varepsilon_0}\|\to 0\), where \(x\) is the solution of equation (29) for \(\delta>0\), \(\varepsilon=\varepsilon_0+i\delta\), \(\delta\to 0\). In addition,

\[ \|(E-\varepsilon A)^{-1}A^n f\|\leq \frac{C}{\varepsilon_0^m}, \tag{38} \]

where \(m\) is an integer, \(m\geq 1\), \(C\) is a constant determined only by the element \(f\) and the operator \(A\), \(n=0,1,2,\ldots\), \(\delta\leq \delta_0(\varepsilon)\). Let \(|\varepsilon|\leq 1\), and let condition (10) be fulfilled. Then the following holds.

Theorem 1.2. Under the conditions listed above, for the solution of equation (29) obtained by the principle of limiting absorption, we have the asymptotic formula

\[ x_{\varepsilon_0}\sim \sum_{n=0}^{\infty}\varepsilon_0^n A^n f. \tag{39} \]

Proof. Let \(x\) be a solution of equation (29), and let \(n\) be any integer, \(n>0\). From the identity

\[ x=\sum_{k=0}^{n+m}\varepsilon^k A^k f+(E-\varepsilon A)^{-1}\varepsilon^{n+m+1}A^{n+m+1}f \tag{40} \]

and condition (38), fixing \(n\) and passing to the limit as \(\delta\to 0\), we obtain

\[ \left\|x_{\varepsilon_0}-\sum_{k=0}^{n}\varepsilon_0^k A^k f\right\| \leq |\varepsilon_0|^{n+1}\left(C+\left\|A^{n+1}\left(\sum_{k=0}^{m-1}\varepsilon_0^k A^k f\right)\right\|\right). \tag{41} \]

Since condition (10) is fulfilled, Theorem 1.2 is proved.

Let in equation (29) \(\varepsilon\to 0\), and suppose that for some integer \(m\geq 1\) the condition

\[ \alpha|\varepsilon_0|^{m+1}<\delta<\beta|\varepsilon_0|, \tag{42} \]

is fulfilled, where \(\alpha>0\), \(\beta>0\), \(|\varepsilon_0|<|\varepsilon|\leq 1\). Denote the solution of equation (29) by \(x(\varepsilon)\).

Theorem 1.3. If conditions (10) and (42) are fulfilled, then for the solution \(x(\varepsilon)\) of equation (29) the asymptotic expansion

\[ x(\varepsilon)\sim \sum_{n=0}^{\infty}\varepsilon^n A^n f \tag{43} \]

is valid.

Proof. From identity (40), for any fixed \(n\) we have the estimate

\[ \left\|x(\varepsilon)-\sum_{k=0}^{n}\varepsilon^{k}A^{k}f\right\|\leq \]

\[ \leq |\varepsilon|^{n+1}\left\|A^{n+1}\left(\sum_{k=0}^{m-1}\varepsilon^{k}A^{k}f\right)+(E-\varepsilon A)^{-1}\varepsilon^{m}A^{n+m+1}f\right\| . \tag{44} \]

As is known,

\[ \left\|\varepsilon(E-\varepsilon A)^{-1}h\right\|\leq \frac{\|h\|}{|\operatorname{Im}\varepsilon_1|} \tag{45} \]

for any \(h\in H\) and \(\varepsilon_1=\dfrac{1}{\varepsilon}\). From condition (42) we obtain that \(|\operatorname{Im}\varepsilon_1|>\gamma|\varepsilon|^{m-1}\) for some \(\gamma\). Hence, from inequality (45) with \(h=A^{m}f\), we finally obtain

\[ \left\|x(\varepsilon)-\sum_{k=0}^{n}\varepsilon^{k}A^{k}f\right\|\leq |\varepsilon|^{n+1}\left(m+\frac{1}{\gamma}\right)K . \]

The theorem is proved.

§ 3. On an Interpolation Problem

Consider the ill-posed problem of recovering an analytic function \(F(z)\) from its values \(F(n)\) at the points \(z=n,\ n=1,2,\ldots\). In what follows it will be shown that this problem also reduces to the solution of an equation of the form (1), for which zero is a spectral point. We give estimates characterizing the stability of the problem. Consider a function \(F(z)\), analytic for \(\operatorname{Re} z>0\), such that

\[ |F(iy)|\leq M\exp((\pi-\varepsilon)|y|),\quad 0<\varepsilon<\pi, \tag{46} \]

\[ |F(z)|\leq M\exp(B|z|). \tag{47} \]

Let us estimate the function \(F(z)\) on any segment \(|y|\leq l\) of the line \(\operatorname{Re} z=\dfrac{1}{2}\), knowing an estimate for \(F(n)\), \(n\geq 1\). Then already with the aid of (46) and (47) one can estimate \(F(z)\) in any bounded domain of the right half-plane (see [1], p. 31).

Theorem 1.4. Suppose that the estimates (46), (47) hold and \(|F(n)|\leq m\), \(n\geq 1\), \(m>0\). Then for \(|y|\leq l\), \(\operatorname{Re} z=\dfrac{1}{2}\), the inequality

\[ |F(z)|\leq \Phi(M,\varepsilon,m) \tag{48} \]

holds, where \(\Phi(M,\varepsilon,m)\to 0\) for fixed \(M,\varepsilon\) and \(m\to 0\).

Proof. To estimate \(F(z)\), we use the formula from [5], p. 149:

\[ F(z)=-\frac{\sin \pi z}{\pi z}\int_{-\infty}^{\infty}P(\xi)\exp(-z\xi)\,d\xi, \tag{49} \]

where

\[ P(z)=-\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{F(\zeta)}{\Phi(\zeta)}\exp(z\zeta)\,d\zeta, \tag{50} \]

\[ \Phi(\zeta)=\frac{\sin \pi \zeta}{\pi \zeta}, \]
and, for sufficiently large \(\operatorname{Re} z<0\),

\[ P(z)=\sum_{n=1}^{\infty}(-1)^n nF(n)\exp(nz). \tag{51} \]

Let us estimate \(|P(\xi)|\) for \(-\infty<\xi<\infty\). For \(\xi\leq a<0\), where \(a\) is arbitrary but fixed, we have

\[ |P(\xi)|\leq m\sum_{n=1}^{\infty} n\exp(n\xi)\leq \frac{m\exp \xi}{(1-\exp a)^2}. \tag{52} \]

To estimate \(|P(\xi)|\) for \(\xi>a\), we use an argument of M. M. Lavrent'ev (see [1], p. 31, formula (79)), from which it follows that

\[ |P(z)|\leq P^{1-\omega}Q^\omega, \tag{53} \]

where
\[ P=\max_{\substack{\operatorname{Re} z=a\\ |y|\leq \varepsilon_1}} |P(z)|,\qquad Q=\max_{|y|\leq \varepsilon/2} |P(z)|, \]
and \(\omega\) is a function harmonic for \(x=\operatorname{Re} z>a\), \(|y|\leq \varepsilon_1\), with \(\omega=1\) for \(x=a\), \(\omega=0\) for \(y=\pm i\varepsilon_1\); \(\varepsilon_1\) is an arbitrary number from the interval \((0,\varepsilon)\). Let \(a=-2\ln 2\). Then from (52) it follows that \(|P(\xi)|\leq m\) for \(\xi\leq a\), and

\[ P\leq m. \tag{54} \]

To estimate \(Q\), we use formula (50) and inequality (46), whence it is not difficult to obtain that, for \(|y|\leq \varepsilon_1\),

\[ Q\leq \max_{|y|\leq \varepsilon_1}|P(z)|<MJ(\varepsilon,\varepsilon_1), \tag{55} \]

where

\[ J(\varepsilon,\varepsilon_1)=\int_0^\infty \frac{x\exp((\pi-\delta)x)\,dx}{\operatorname{sh}\pi x} = O\left(\frac{1}{\omega^2}\right),\qquad \delta=\varepsilon-\varepsilon_1. \]

From estimate (52) we obtain

\[ \left| \int_{-\infty}^{a} P(\xi)\exp\left(-\left(\frac12+iy\right)\xi\right)\,d\xi \right| \leq 2m. \tag{56} \]

From estimate (55) it follows that, for any \(N>a\),

\[ \int_N^\infty P(\xi)\exp\left(-\left(\frac12+iy\right)\xi\right)\,d\xi\to 0 \tag{57} \]

for fixed \(M\) and \(m\to 0\), while from estimates (53), (54) it follows that also

\[ \int_a^N P(\xi)\exp\left(-\left(\frac12+iy\right)\xi\right)\,d\xi\to 0 \tag{58} \]

for fixed \(N, M\) and \(m\to 0\). From (57) and (58) it follows that

\[ \int_a^\infty P(\xi)\exp\left(-\left(\frac12+iy\right)\xi\right)\,d\xi\to 0, \]
if \(M\) is bounded and \(m\to 0\). Then from formula (49) and estimates (56)—(58) the assertion of the theorem follows.

Let us note that for the function \(\Phi\), from inequality (48) an explicit expression can be written on the basis of (52)—(58).

We record an a priori estimate in the problem of finding the function \(F(z)\). From formula (50) it follows that, for real \(z\),

\[ P(z)=-\frac{1}{2\pi}\int_{-\infty}^{\infty} g(t)\exp(itz)\,dt, \tag{59} \]

where

\[ g(t)=\frac{\pi t F(it)}{\operatorname{sh}\pi t}. \tag{60} \]

From formula (59) and the theory of the Fourier integral it follows that

\[ \|g\|=\sqrt{2\pi}\,\|P\|, \tag{61} \]

where the norm is taken in the sense of \(L_2(-\infty,\infty)\). Equality (61) should be regarded as an unimprovable a priori estimate of the required function \(F(z)\) on the imaginary axis in problems which reduce to the interpolation problem considered above.

Let us derive the singular equation to which the problem of finding the function \(F(z)\) reduces. For simplicity we shall assume that

\[ |F(n)|<q^n, \tag{62} \]

where \(0<q<1\). (The general case is reduced to this by replacing \(F(z)\) by \(F_1(z)=F\exp(-2Bz)\).) Introduce the function \(H(t)\)

\[ H(t)=\sum_{n=1}^{\infty}\frac{n c_n(-1)^{n+1}}{n-it}, \tag{63} \]

where \(c_n=F(n)\), \(-\infty<t<+\infty\). Condition (62) ensures the existence of \(H(t)\). Under assumptions (46) there exists the Cauchy integral \(K(g)\):

\[ K(g)=\frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{g(\xi)\,d\xi}{\xi-t}. \tag{64} \]

We shall prove that

\[ g-K(g)=H. \tag{65} \]

Consider the formula of work [5] on p. 152. Correcting in the left-hand side the plus sign to a minus, putting \(\Phi=\sin \pi z/\pi z\) and \(q=0\) in the right-hand side, we obtain

\[ \frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{f(\zeta)\,d\zeta}{\zeta-z} = -f(z)+\sum_{n=1}^{\infty}\frac{n c_n(-1)^n}{z-n}, \tag{66} \]

where \(\operatorname{Re} z>0\), \(f(z)=F(z)\pi z/\sin\pi z\). Letting, in equality (66), the point \(z\) tend to the point \(it\), with \(t\) real, using the Sokhotski formulas and passing to the notation (60) and (63), we obtain equation (65). We note that it belongs to the type of equations on the spectrum.

§ 4. On A Priori Estimates of Solutions

Let us consider the question of obtaining an a priori estimate for the solution of equation (1) and of the more general equation

\[ Ax-\lambda x=f. \tag{67} \]

Estimates of solutions of equation (67) can sometimes be obtained by constructing an operator \(B\) such that

\[ (Ax-\lambda x,\;Bx)>0. \tag{68} \]

It is easy to construct a nonlinear operator \(B\) such that inequality (68) is satisfied for all \(x\) that are not eigenvectors of the operator \(A\). Indeed, if we take

\[ Bx=\|x\|^{2}Ax-(Ax,x)x, \]

then \((Bx,x)=0\), and (68) becomes the inequality

\[ (Ax,Bx)=\|x\|^{2}\|Ax\|^{2}-(Ax,x)^{2}>0, \]

which is satisfied if \(Ax\ne \alpha x\), for any \(\alpha\). We shall seek an operator \(B\) satisfying (68) in the following form:

\[ Bx=A(Cx)-C(Ax), \tag{69} \]

where \(C=C^{*}\), \(R(C)\subset D(A)\), \(R(A)\subset D(C)\), if \(R(C)\), \(R(A)\) are the ranges of the operators \(C\) and \(A\); the operators \(A\) and \(C\) do not commute. Then \((Bx,x)=0\), and inequality (68) takes the form

\[ (Ax,Bx)>0. \tag{70} \]

Examples show that for differential operators of elliptic type, as the operator \(C\) one may take the operator of multiplication by a suitable function \(\varphi\).

Consider the case of the Helmholtz operator. Let \(\Omega\) be a domain of \(n\)-dimensional space with points \(x=(x_{1},\ldots,x_{n})\). Let the boundary \(\Gamma\) of the domain \(\Omega\) be an unbounded smooth surface, star-shaped with respect to the origin, and let \(\Omega\) be located outside \(\Gamma\). Consider in \(\Omega\) sufficiently smooth functions \(u(x)\) that decrease sufficiently rapidly at infinity, with

\[ u|_{\Gamma}=0. \tag{71} \]

Let

\[ Lu=\Delta u+\lambda u, \]

where \(\Delta\) is the Laplace operator, \(\lambda\) is real, \(\lambda>0\). In formula (69) take \(A=\Delta\), \(Cu=\varphi u\); then

\[ Bu=2(\nabla\varphi,\nabla u)+u\Delta\varphi, \tag{72} \]

where \(\nabla\varphi=(\varphi_{1},\ldots,\varphi_{n})\), \(\varphi_i=\dfrac{\partial\varphi}{\partial x_i}\), \(i=1,2,\ldots,n\). We shall show that, for a broad class of functions, the condition

\[ (Lu,Bu)>0, \tag{72'} \]

is satisfied, where the scalar product is taken in \(L_{2}(\Omega)\). We prove the identity

\[ (Lu,Bu)=\frac{1}{2}\int_{\Omega} u^2\Delta^2\varphi\,dx -2\int_{\Omega}\sum_{i,k=1}^{n}\varphi_{ik}u_i u_k\,dx +\int_{\Gamma}(\nabla\varphi,\nu)(\nabla u)^2\,d\sigma, \tag{73} \]

where \(\nu=(a_1,\ldots,a_n)\) is the outward normal to \(\Gamma\). From the identity (73), (72′) follows if the function \(\varphi\) satisfies the conditions

\[ \Delta^2\varphi\geqslant 0,\qquad (\nabla\varphi,\nu)>0,\qquad \sum_{i,k}\varphi_{ik}\lambda_i\lambda_k<0 \tag{74} \]

for any \(\lambda_1,\ldots,\lambda_n\) not all equal to zero. As functions \(\varphi\) one may take \(\varphi=-|x|^2\) for any \(n\), \(\varphi=-|x|^3\) for \(n>3\), and other functions. The inequality \((\nabla\varphi,\nu)>0\) then follows from the star-shapedness condition for \(\Gamma\). We note that the identities encountered in the theory of Helmholtz’s equation, for example see [14–16], are particular cases of (73). Verification of the identity (73) presents no difficulties; one need only take into account that, if \(u|_{\Gamma}=0\), then \(u_i a_k=u_k a_i,\ i,k=1,2,\ldots,n\), and

\[ \cos(\widehat{\nu,\nabla\varphi}) = \cos(\widehat{\nabla\varphi,\nabla u})\cos(\widehat{\nabla u,\nu}). \]

From the identity (73) one can obtain a priori estimates for the solution of the equation

\[ Lu=f. \tag{75} \]

in the class of functions \(u(x)\) sufficiently smooth and sufficiently rapidly decreasing at infinity. Indeed, from (75) it follows that

\[ \lambda\int_{\Omega}u^2dx = \int_{\Omega}fudx+\int_{\Omega}(\nabla u)^2dx. \tag{76} \]

Putting \(\varphi=-|x|^2\) in [73], from (76) we obtain that

\[ \lambda\int_{\Omega}u^2dx +\int_{\Omega}(\nabla u)^2dx +\int_{\Gamma}|(x,\nu)|(\nabla u)^2\,d\sigma = \]

\[ = \int_{\Omega}fudx-\frac{1}{2}\int_{\Omega}uBf\,dx, \]

whence it follows, for example, that for \(\lambda>0\)

\[ \|u\|\leqslant \frac{1}{\lambda}\|g\|, \]

where \(g=(n+1)f+2rf_r,\ r=|x|\). Similar, but less general, results can be obtained for the operators \(A=\Delta^2\), \(A=\Delta+c(x)\), and also for operators with variable coefficients differing little from constants.

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Received by the editors
June 21, 1965