UDC 517.948.35:517.946.9

ILL-POSED PROBLEMS IN HILBERT SPACE AND STABLE METHODS FOR THEIR SOLUTION

Yu. T. Antokhin

The present work is a continuation of the author’s papers [1, a)—c)] and is devoted to the study of ill-posed problems for linear self-adjoint equations in a Hilbert space \(H\).

Let \(A\) and \(B\) be given linear, unbounded operators with domains \(D(A)\) and \(D(B)\) everywhere dense in \(H\).

Problem I. Find the solution of the equation

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

where \(f\) is given and \(x\) is the required element of the space \(H\); equation (1) is solvable, and zero is a point of the continuous spectrum of the operator \(A\).

Problem II. Find the element \(x\), if

\[ x=Bf, \tag{2} \]

where \(f\) is a given element of the space \(H\), \(f \in D(B)\).

Let us explain the formulation of Problems I and II. If, as usual, a well-posed problem in \(H\) is understood to be one for which the element \(x \in H\) always exists, is unique, and depends continuously (in the norm of \(H\)) on the given element \(f \in H\), then Problems I and II are ill-posed. Indeed, in Problems I and II the conditions of existence and uniqueness are satisfied, but there is no continuous dependence on the given element, since in the first case there exists a sequence \(\{x_n\}\) \((n=1,2,\ldots)\) such that \(\|Ax_n\|\to 0\), \(\|x_n\|=1\), while in the second case there exists a sequence \(\{f_n\}\) for which \(\|Bf_n\|\to \infty\), whereas \(\|f_n\|\to 1\) as \(n\to\infty\). Obviously, the converse assertion is also true, i.e., if a linear ill-posed problem of the form (1) or (2) is solvable and, moreover, in a unique way, then in equation (1) zero is a point of the continuous spectrum of the operator \(A\), and in equation (2) the operator \(B\) is unbounded.

We note that there is an extensive and important class of ill-posed problems in optimal control theory where an explicit specification of the operators \(A\) or \(B\) is difficult; for example, see [9, 16, 28]. Stable methods for solving some problems of this type are presented in the work of A. N. Tikhonov [3, c)].

Most works on ill-posed problems are devoted to equation (1). Here the most highly developed methods are those set forth in the works of M. M. Lavrent’ev [2, a)], V. K. Ivanov [4, a)—c)] and A. N. Tikhonov [3, a)—d)]. In these works the operator \(A\) (in the linear case) is usually assumed to be continuous or even completely continuous (see also [12, 13, 17]). The requirement of continuity of the operator \(A\) (boundedness) is removed in Problem I, which is studied in the present work. This extends the range of application of the results obtained, in particular to the theory of differential equations (see [1, a)—c)]).

Problems I and II are studied below in the case where \(A\) and \(B\) are self-adjoint or positive self-adjoint operators.

The meaning of the terms “to find a solution” and “to find an element” is explained in Definition 1 of § 1. It consists in constructing a method, “stable” with respect to small changes of \(f\), for solving problems I and II. For this purpose a family of bounded linear operators \(\{R_\varepsilon\}\), \(0<\varepsilon<1\), is introduced such that \(R_\varepsilon f\to x\) as \(\varepsilon\to 0\) for any solution \(x\) of problems I and II. Each such family (a regularizer) gives rise to several stable methods (Theorem 2.1). For this reason, the construction of regularizers is considered separately from questions of constructing stable methods. Regularizers are constructed for the case of self-adjoint operators \(A\) and \(B\) (Theorem 1.1), and also for those non-self-adjoint \(A\) and \(B\) for which a self-adjoint operator \(\Delta\) is explicitly specified such that \(\|B\Delta^{-1}\|<\infty\), \(\|(\Delta A)^{-1}\|<\infty\) (see Theorem 1.2).

In § 2 stable methods are constructed and examples illustrating theorems of a general character are given. It is shown that, for the convergence of approximate methods, there is no need to use any a priori information about the solution, except for the fact of its existence.

In § 3 the methods set out above are used to study that important case of problems I and II in which the solution \(x\) belongs to some linear manifold \(L\) of the original space \(H\). Here the principal new requirement on the constructed regularizers is that \(R_\varepsilon f\in L\) for every \(f\in H\). This is achieved either by introducing a new Hilbert space \(\dot H\) and reducing the problems of § 3 to problems of types I and II in the space \(\dot H\), or by applying to the given element \(f\) a suitable smoothing procedure, for example by applying the operators \(A^{-1}\) and \(B\) to the element \(\exp(-\varepsilon\Delta)f\), where \(\Delta\) is a certain positive definite operator, \(\Delta=\Delta^*\). The developed method is applied to the solution of integral equations of the first kind when \(H=L_2(-\infty,+\infty)\) and \(L\) is the set of functions with bounded derivatives of higher order.

The main result of the paper is presented in § 4, devoted to estimates of the rate of convergence of approximate methods. In Theorem 4.1 a sufficiently broad class \(R\) of regularizers \(R_\varepsilon\) of problem I is introduced. For \(R_\varepsilon\in R\) there exists \(R_\varepsilon^{-1}\), \(\|R_\varepsilon\|=\varepsilon^{-1}\), and, almost always, the operator \((A^{-1}-R_\varepsilon)^{-1}\) exists. Then for any \(R_\varepsilon\in R\),
\(\|x-R_\varepsilon f\|>q^{-1/\varepsilon}\) for some \(q>1\), for any solution \(x\) of problem I, where now the operator \(A\) is positive, self-adjoint, and \(\|A\|<1\). At the same time, there exist regularizers \(R_\varepsilon\in R\) such that, with some constant \(c>0\),

\[ \|x-R_\varepsilon f\|\leq c\varepsilon^n \tag{*} \]

for any \(n>0\). The last estimate is obtained under the assumption that the solution \(x\) is sourcewise representable, i.e. \(x=A^n y\) for some \(y\in H\). The condition of sourcewise representability was introduced in the author’s papers [1, a)—b)]. It makes it possible to obtain effective estimates of the convergence of various approximate methods also in the case when the solution does not belong to compact sets (see § 4). The meaning of the requirement \(x=A^n y\) is explained by some examples in the author’s paper [1, c)] and in §§ 2, 3 of the present paper. Estimate (*) is an analogue of increased-accuracy estimates for the problem of differentiation, the usefulness of which in problems of analytic continuation was experimentally discovered in the well-known works of P. Garabedian [33]. In addition, it is proved that, when regularizers \(R_\varepsilon\in R\) are used, errors in specifying \(f\) are always dangerous to some extent, and for some \(f\) they are extremely dangerous, i.e.

\[ \|x-R_\varepsilon f_\delta\|\asymp \|x-R_\varepsilon f\|+\frac{\delta}{\varepsilon} \]

for some \(f\) and \(f_\delta\), \(\|f-f_\delta\|\leq \delta\).

§ 1. REGULARIZERS OF ILL-POSED PROBLEMS

In §§ 1 and 2 a construction is given of stable methods for solving problems I and II.

We shall give a definition of a method for solving problems I and II that is stable with respect to small changes of the prescribed element \(f\).

Let \(x\) be the solution of problem I or II.

Definition 1. We shall say that a method for solving problems I and II has been proposed which is stable with respect to small changes of the element \(f\), if, for every \(\delta\), \(0<\delta<1\), there is specified a linear bounded operator \(\widetilde R_\delta\) such that

\[ x=\lim_{\delta\to 0}\widetilde R_\delta f_\delta, \tag{3} \]

where \(f_\delta\) is an arbitrary element of \(H\) satisfying the condition

\[ \|f-f_\delta\|\leq \delta . \tag{4} \]

Definition 2. By a regularizer of problems I and II we shall mean any family of linear bounded operators \(R_\varepsilon\), \(0<\varepsilon<1\), such that

\[ x=\lim_{\varepsilon\to 0} R_\varepsilon f . \tag{5} \]

Definition 1 is a concretization of the concept of a stable algorithm introduced by A. N. Tikhonov in [3, b], and the family of elements \(\widetilde R_\delta f_\delta\) is analogous to the “regularized family of approximate solutions of problem I” from the works of A. N. Tikhonov [3, a), c)].

We shall prove a theorem on the existence of regularizers and apply it in § 2 to the construction of stable methods for solving the basic problems I and II. In what follows, in formula (5), instead of \(R_\varepsilon\) we shall also write \(R_\varepsilon(A)\), \(R_\varepsilon(B)\), depending on which problem (I or II) is under consideration. We shall also call \(R_\varepsilon(A)\) and \(R_\varepsilon(B)\) regularizers of the operators \(A\) and \(B\).

Theorem 1.1. If in problems I and II the operators \(A\) and \(B\) are self-adjoint, then the regularizers \(R_\varepsilon(A)\) and \(R_\varepsilon(B)\) exist.

Proof. Consider problem I. If \(x\) is the solution of this problem, then

\[ x=\int_{-\infty}^{+\infty}\frac{1}{\lambda}\,dE_\lambda f, \tag{6} \]

where \(\{E_\lambda\}\) is the resolution of the identity for the operator \(A\). Take a real function \(\rho=\rho(\varepsilon,\lambda)\), \(0<\varepsilon\leq 1\), \(-\infty<\lambda<\infty\), satisfying the following conditions:

\(1^\circ)\) \(\rho\) is continuous in \(\lambda\) for each fixed \(\varepsilon\) and is bounded for \(|\lambda|\leq \infty\), \(|\rho(\varepsilon,\lambda)|\leq 1\);

\(2^\circ)\) \(\rho/b(\varepsilon)\lambda^{a(\varepsilon)}\to 1\) as \(\lambda\to 0\) for some continuous functions \(a(\varepsilon)\) and \(b(\varepsilon)\), \(a(\varepsilon)\geq 1\);

\(3^\circ)\) \(\rho\to 1\) as \(\varepsilon\to 0\) uniformly in \(\lambda\) in every interval of the form \((-\infty,-\eta)\) or \((\eta,+\infty)\), where \(\eta\) is an arbitrary number, \(\eta>0\).

Consider the operator \(R_\varepsilon(A)\) defined by the formula

\[ R_\varepsilon(A)f=\int_{-\infty}^{+\infty}\frac{1}{\lambda}\rho(\varepsilon,\lambda)\,dE_\lambda f. \tag{7} \]

The operator (7) is bounded, and its norm \(\|R_\varepsilon\|\) is given by the formula

\[ \|R_\varepsilon\|=\max_{-\infty<\lambda<+\infty}\left|\frac{1}{\lambda}\rho(\varepsilon,\lambda)\right|. \tag{8} \]

Let us prove that the operator (7) is indeed a regularizer of problem I, i.e., \(R_\varepsilon f\to x\) as \(\varepsilon\to0\). For any \(\eta>0\) we have:

\[ x-R_\varepsilon f = \int_{|\lambda|<\eta}\frac{1-\rho}{\lambda}\,dE_\lambda f + \int_{|\lambda|>\eta}\frac{1-\rho}{\lambda}\,dE_\lambda f = J_1+J_2. \tag{9} \]

In equality (9), the integral \(J_1\) exists for any \(\varepsilon\) and \(\eta\), and \(\|J_1\|\to0\) as \(\eta\to0\) uniformly in \(\varepsilon\), since the integral (6) exists. For any fixed \(\eta>0\), as \(\varepsilon\to0\) the integral \(J_2\to0\) by virtue of \(3^\circ\). Therefore \(x-R_\varepsilon f\to0\) as \(\varepsilon\to0\), and the operator (7) is the regularizer \(R_\varepsilon(A)\) of problem I.

Let us construct a regularizer of problem II. If \(\{E_\lambda\}\) is the resolution of the identity for the operator \(B\), and \(x\) is the solution of problem II, then

\[ x=\int_{-\infty}^{+\infty}\lambda\,dE_\lambda f. \tag{10} \]

Consider the operator

\[ R_\varepsilon(B)f=\int_{-\infty}^{+\infty}\lambda\varphi(\varepsilon,\lambda)\,dE_\lambda f, \tag{11} \]

where \(\varphi=\varphi(\varepsilon,\lambda)\) is continuous in \(\lambda\), \(|\varphi|\leqslant1\) for \(0<\varepsilon\leqslant1\), \(-\infty<\lambda<\infty\), \(\varphi\to1\) as \(\varepsilon\to0\) uniformly in \(\lambda\) on every interval \((-\eta,\eta)\), \(\eta>0\), and

\[ \max_{-\infty<\lambda<\infty}|\lambda\varphi|<\infty. \]

As above, we find that

\[ \|R_\varepsilon(B)\|=\max_{-\infty<\lambda<\infty}|\lambda\varphi|, \tag{12} \]

and \(Bf=\lim_{\varepsilon\to0}R_\varepsilon f\). Therefore \(R_\varepsilon(B)\) is a regularizer of problem II.

The theorem is completely proved.

Remark. From formulas (8) and (12) it is clear that the norms of the regularizers \(R_\varepsilon(A)\) and \(R_\varepsilon(B)\) are estimated solely as functions of the analytic structure of the functions \(\rho(\varepsilon,\lambda)\) and \(\varphi(\varepsilon,\lambda)\). This is important in applications (see below the remark to Theorem 2.1). At the same time, the equality \(x=\lim R_\varepsilon f\) is a consequence of the fact of existence of solutions of problems I and II. The order of the infinitesimal quantity \(\|x-R_\varepsilon f\|\) can be estimated only under additional assumptions of various kinds. For example, in the author’s papers [1, a), b)] this is done for “source-representable” solutions of problems I and II, i.e., when \(x=Ay\) for some \(y\in H\), \(\|y\|\leqslant1\) (problem I), or \(\|B^2 f\|\leqslant1\) (problem II).

As examples of the choice of the functions \(\rho\) and \(\varphi\), let us consider the following cases of problems I and II.

\(1^\circ\) In problem I the operator \(A\) is positive. Then one may take \(\rho=\lambda/(\lambda+\varepsilon)\),

\[ R_\varepsilon(A)=(\varepsilon E+A)^{-1}, \]

and if \(x_\varepsilon=R_\varepsilon(A)f\), then \(x=\lim_{\varepsilon\to0}x_\varepsilon\), where

\[ \varepsilon x_\varepsilon+Ax_\varepsilon=f. \]

This case was studied in the author’s papers [1, a), b)], and in special cases in the papers [11—13, 17].

\(2^\circ)\) If in problem I the operator \(A\) is an arbitrary self-adjoint operator, then one may take \(\rho=\lambda/(\lambda+i\varepsilon)\), \(\varepsilon>0\). If we denote \(x_\varepsilon=R_\varepsilon f\), then \(x=\lim_{\varepsilon\to0}x_\varepsilon\), where

\[ i\varepsilon x_\varepsilon+Ax_\varepsilon=f. \]

In the present case we solve problem I by a method analogous to the “principle of limiting absorption” for the Helmholtz equation (see [14, 6]). For a completely continuous operator \(A\) this method was considered in [11]. It can be applied to the Dirichlet problem for the wave equation \(x_{t^2}-x_{s^2}=f(t,s)\), where \(x(t,s)\) and \(f(t,s)\) are functions defined in a certain domain \(G\) of the \((t,s)\)-plane. As the operator \(A\) one may take the operators \(\partial^4/\partial t^2\partial s^2\), \(\partial^4/\partial t^4-\partial^4/\partial s^4\), etc. The differential operators mentioned are self-adjoint in spaces of functions obtained from finite ones by completion in the norm \(W_2^l\) for \(l=2\), etc.

\(3^\circ)\) If in problem II the operator \(B\) is positive, then we take \(\varphi_1=\exp(-\varepsilon\lambda)\). If \(u=u(\varepsilon)=R_\varepsilon(B)f\), then \(x\equiv Bf=\lim_{\varepsilon\to0}u(\varepsilon)\), where

\[ \frac{du}{d\varepsilon}=-Bu,\qquad u(0)=f. \]

The regularizer \(R_\varepsilon(B)=B\exp(-\varepsilon B)\) was studied in the author’s work [1, c)]. If in problem II the operator \(B\) is an arbitrary self-adjoint operator, then we take \(\varphi_2=\exp(-\varepsilon\lambda^2)\). In this case \(Bf=\lim_{\varepsilon\to0}v(\varepsilon)\), where \(dv/d\varepsilon=-B^2v,\ v(0)=f\).

Such regularizers of the operator \(B\) are analogous to the “damping factor” method in the theory of divergent integrals and find application (see V. K. Ivanov’s work [4, b]) to the Cauchy problem for the Laplace equation). In applications a useful method is that in which \(R_\varepsilon(B)=B(E+\varepsilon B^2)^{-1}\). In this case, if \(w_\varepsilon=R_\varepsilon(B)f\), then \(Bf=\lim_{\varepsilon\to0}w_\varepsilon\), where \(w_\varepsilon\) satisfies the equation

\[ \varepsilon B^2 w_\varepsilon+w_\varepsilon=Bf, \]

which is solvable for every \(f\in H\), since \(\|B(E+\varepsilon B^2)^{-1}\|<\infty\).

\(4^\circ)\) In the case when \(H\) is some space of functions \(\{f\}\), and \(Bf\) is represented by a divergent (on the class of all functions \(f\)) integral, it is possible to compute explicitly and to study in detail the following regularizer of problem II:

\[ R_\varepsilon(B)f=\int_{|\lambda|\le 1/\varepsilon}\lambda\,dE_\lambda f. \tag{13} \]

For example, if \(f=f(t)\), \(f\in H=L_2(-\infty,+\infty)\), then \((Bf=f^{(n)}(t))\)

\[ Bf=\varkappa\int_{-\infty}^{+\infty}s^n\exp(-its)\tilde f(s)\,ds, \]

\[ R_\varepsilon(B)f=\varkappa\int_{|s|\le 1/\varepsilon}s^n\exp(-its)\tilde f(s)\,ds, \tag{14} \]

where \(\varkappa=(-i)^n/2\pi\); \(\tilde f(s)\) is the Fourier transform of the function \(f(t)\),

\[ \tilde f(s)=\int_{-\infty}^{+\infty} f(t)\exp(its)\,dt. \]

For the regularizer (13)

\[ \|R_\varepsilon(B)\|=\frac{1}{\varepsilon}. \tag{15} \]

In the author's paper [1, a), b)], instead of the method used in Theorem 1.1, ideas from the theory of interpolation of entire functions were used for constructing approximate solutions of problem I. This is equivalent to approximating the integrand in (6) by polynomials. The role of the parameter \(\varepsilon\) is then played by the degree of the polynomial.

Let us show how to apply Theorem 1.1 to the case when the operators \(A\) and \(B\) in problems I and II are not self-adjoint. Introduce an auxiliary operator \(\Delta\), which will often be used below. The linear operator \(\Delta\) has the following properties: a) \(\Delta=\Delta^*\), \(D(\Delta)\) is everywhere dense in \(H\); b) there exists a bounded inverse \(\Delta^{-1}\); c) for all \(g\in D(\Delta)\) one always has \((\Delta g,g)>k\|g\|^2\), where \(k\) is some constant.

The last condition is not essential for the following Theorem 1.2 and will be used only in § 3.

Theorem 1.2. Suppose that in problems I and II the following additional conditions are satisfied. There exists an operator \(\Delta\) such that

\(1^\circ)\) \(f\in D(\Delta)\);

\(2^\circ)\) for the operator \(\Delta A\) there exists a bounded inverse;

\(3^\circ)\) the operator \(B\) is such that \(\|B\Delta^{-1}\|<\infty\);

\(4^\circ)\) the operator \(\Delta\) satisfies the properties a)—c) listed above.

Then for problems I and II there exist regularizers.

Proof. Consider problem I. Since the operator \(\Delta\) is self-adjoint, by Theorem 1.1 it has a regularizer; let \(R_\varepsilon(\Delta)\) be one of these regularizers. Denote by \(R_\varepsilon(A)\) the operator

\[ R_\varepsilon(A)=(\Delta A)^{-1}R_\varepsilon(\Delta) \tag{16} \]

and prove that (16) is a regularizer for problem I. Let \(x_\varepsilon=(\Delta A)^{-1}R_\varepsilon(\Delta)f\). By the hypothesis of the theorem \(f\in D(\Delta)\), and therefore \(x-x_\varepsilon\in D(\Delta A)\) and \(\Delta A(x-x_\varepsilon)=\Delta f-R_\varepsilon(\Delta)f\to0\) as \(\varepsilon\to0\). Since \((\Delta A)^{-1}\) exists, it follows that \(x_\varepsilon\to x\) as \(\varepsilon\to0\).

For problem II consider the operator \(R_\varepsilon(B)=B\Delta^{-1}R_\varepsilon(\Delta)\) and prove that it serves as a regularizer for \(B\). Indeed, since \(f\in D(\Delta)\cap D(B)\), we have \(Bf-R_\varepsilon(B)f=B\Delta^{-1}(\Delta f-R_\varepsilon(\Delta)f)\to0\) as \(\varepsilon\to0\), because \(\|B\Delta^{-1}\|<\infty\) and \(R_\varepsilon(\Delta)f\to\Delta f\) by the definition of \(R_\varepsilon(\Delta)^*)\).

§ 2. STABLE METHODS FOR SOLVING PROBLEMS

Let \(R_\varepsilon\) be a regularizer of problem I or II, \(\|R_\varepsilon\|\le N_\varepsilon\), \(N_\varepsilon\to\infty\) as \(\varepsilon\to0\), \(N_\varepsilon=1\) for \(\varepsilon=1\), and \(N_\varepsilon\) a continuous and monotone function of \(\varepsilon\).

Theorem 2.1. Every regularizer \(R_\varepsilon\) of problem I or II generates a stable method of solution by the formula

\[ x=\lim_{\delta\to0}\widetilde R_\delta f_\delta, \tag{17} \]

where \(\|f-f_\delta\|\le\delta\le1\), \(\widetilde R_\delta=R_\varepsilon\) for a certain choice \(\varepsilon=\varepsilon(\delta)\).

Proof. We apply the John–Lavrentiev method. Let \(R_\varepsilon\) be any regularizer, \(\omega\) an arbitrary number from the interval \((0,1)\), and let \(\varepsilon_\delta\) be the root of the equation

\[ N_\varepsilon=\delta^{-\omega}, \tag{18} \]

\(\varepsilon_\delta\to0\) as \(\delta\to0\). Since \(\|R_\varepsilon(f-f_\delta)\|\le\delta N_\varepsilon\), \(R_\varepsilon f\to x\) as \(\varepsilon\to0\), the theorem is proved for \(\varepsilon(\delta)=\varepsilon_\delta\), which follows from the inequality

\(^*)\) Let us explain Theorem 1.2. If the passage from \(f\) to \(\bar f_\varepsilon=\Delta^{-1}R_\varepsilon(\Delta)f\) is called a smoothing operation, then the theorem asserts, in particular, that when \(\|B\Delta^{-1}\|<\infty\) one and the same smoothing gives both \(R_\varepsilon(\Delta)\) and \(R_\varepsilon(B)\): \(\Delta\bar f_\varepsilon=R_\varepsilon(\Delta)f\), \(B\bar f_\varepsilon=R_\varepsilon(B)f\). For example, smoothing \(f(t)\) in the problem \(x=\Delta f=-f''(t)\), \(f\in L_2(-\infty,+\infty)\), we obtain a regularizer in the problem \(x=Bf=-f''(t)+q(t)f(t)\), where \(q(t)\) is a bounded function, \(|q|\ll1\), \(-\infty<t<\infty\).

\[ \|x-R_{\varepsilon_\delta} f_\delta\| \leq \|x-R_{\varepsilon_\delta} f\|+\|R_{\varepsilon_\delta}(f-f_\delta)\|; \]

\[ \|x-R_{\varepsilon_\delta} f_\delta\| \leq \|x-R_{\varepsilon_\delta} f\|+\delta^{1-\omega}. \tag{19} \]

Remark. If, in constructing \(R_\varepsilon\), we use formulas (7) and (11) and the functions \(\rho\) or \(\varphi\) are known to us, then \(N_\varepsilon\) can be estimated by formulas (8) and (12), after which equation (18) can be solved and \(\varepsilon=\varepsilon_\delta\) explicitly indicated, for which \(R_{\varepsilon_\delta} f_\delta \to x\) as \(\delta \to 0\).

Thus, for example, if \(Ax=f,\ A>0,\ A=A^*\), then \(R_\varepsilon=(\varepsilon E+A)^{-1}\), \(N_\varepsilon=\dfrac{1}{\varepsilon}\) for \(\omega=\dfrac{1}{2}\), \(\varepsilon_\delta=\sqrt{\delta}\), and we obtain that \(x=\lim x_\delta\), where

\[ \sqrt{\delta}\,x_\delta+Ax_\delta=f. \]

For the last problem, the possibility of choosing \(\varepsilon\) was connected by M. M. Lavrent’ev with additional a priori restrictions on the sought solution \(x\) of the problem \(Ax=f\) (see [2, a)]).

The noted effectiveness of computing \(\varepsilon_\delta\) from \(\delta\), independently of the properties of \(x\), is available for problems of a special type, for example, in the works [8, 3, b)] and [11], but the nature of this phenomenon is not revealed. Theorem 2.1 says that, having at our disposal only the fact of existence of a solution, we are able to indicate an approximate formula for finding \(x\), \(R_{\varepsilon_\delta} f_\delta \to x\), but in this case the original problem is replaced by a new ill-posed problem of estimating \(\|x-R_{\varepsilon_\delta} f_\delta\|\) in terms of \(\delta\). A priori assumptions, for example, specifying the size of the compact set to which the solution belongs, are needed only if one wishes to determine the order of the infinitesimal quantity \(\|x-R_{\varepsilon_\delta} f_\delta\|\) as a function of \(\delta\).

Let us give examples of applications of Theorem 2.1.

Example 1. Suppose that in problem II the operator \(B\) has the following form:

\[ B=\sum_{n=1}^{\infty} A_n, \tag{20} \]

where \(A_n\) are bounded operators, \(\|A_n\|=a_n\), \(\sum a_n=\infty\), \(a_1=1\). If \(f\in D(B)\), then the series \(\sum A_n f\) converges, and as \(R_\varepsilon(B)\) one may take the partial sum

\[ R_\varepsilon(B)f=\sum_{1<n\leq [1/\varepsilon]} A_n f, \]

where \([1/\varepsilon]\) is the integer part of the number \(1/\varepsilon\), \(0<\varepsilon<1\). As \(N_\varepsilon\) we take any monotone continuous function, \(N_\varepsilon\big|_{\varepsilon=1}=1\), such that

\[ \sum_{1\leq n\leq [1/\varepsilon]} a_n \leq N_\varepsilon . \tag{21} \]

In particular, if \(a_n=1\), then \(N_\varepsilon=\dfrac{1}{\varepsilon}\). Equation (18) has the solution \(\varepsilon_\delta=\delta^\omega\), and we obtain

\[ Bf=\lim_{\delta\to 0}\sum_{1\leq n\leq [\delta^{-\omega}]} A_n f_\delta,\qquad \|f-f_\delta\|\leq \delta . \tag{22} \]

Problem I for \(A=A^*,\ 0<A<E\) reduces to the case \(a_n\leq 1\) (see [1, b)]), when \(A_n=(E-A)^n\).

The idea of using partial sums was employed in the works [2, a), 3, 15].

Example 2. If in problem II \(B=B^*\), \(B>0\), then \(R_\varepsilon(B)=B\exp(-\varepsilon B)\) (see Example \(3^\circ\), § 1), \(\|R_\varepsilon\|\leq 1/(e\varepsilon)\), \(N_\varepsilon=\dfrac{1}{\varepsilon}\), and

\[ Bf=\lim_{\delta\to 0} B\exp(-\delta^\omega B)f_\delta,\qquad \|f-f_\delta\|\leq \delta \]

for any \(\omega\in(0,1)\).

Example 3. Let \(B=\exp(\Delta^\alpha)\), where \(0<\alpha<1\), \(\Delta\) is the operator introduced in § 1. We take

\[ R_\varepsilon(B)=\exp(\Delta^\alpha-\varepsilon\Delta),\quad \|R_\varepsilon\|\leq N_\varepsilon=\exp(\bar\alpha/\varepsilon) \]

(formula (12), where \(\lambda\) is replaced by \(\exp(\lambda^\alpha)\)), \(\bar\alpha=(1-\alpha)\alpha^{\alpha/(1-\alpha)}\), \(\bar\varepsilon=\varepsilon^{\alpha/(1-\alpha)}\).

Let \(\omega\in(0,1)\), \(\delta\in(0,\delta_0)\), where \(\delta_0=\exp(-\bar\alpha/\omega)\). Then, as \(\delta\to 0\), \(\varepsilon_\delta=(-\bar\alpha/(\omega\ln\delta))^{(1-\alpha)/\alpha}\), and for \(\|f-f_\delta\|\leq\delta\leq\delta_0\) we have

\[ \exp(\Delta^\alpha)f=\lim_{\delta\to 0}\exp(\Delta^\alpha-\varepsilon_\delta\Delta)f_\delta . \tag{23} \]

Example 4. Suppose that in problem I the operator \(A=A^*\), \(A>0\). Consider the Cauchy problem for the equation

\[ \frac{du}{dt}=-Au \tag{24} \]

with the initial condition

\[ u(0)=x, \tag{25} \]

where \(x\) is the solution of problem I, \(t\geq 0\). The solution of problem (24), (25) exists and is given by the formula

\[ u(t)=\exp(-tA)x . \tag{26} \]

It follows from formula (26) that \(u(t)\to 0\) as \(t\to\infty\); hence \(x=\lim_{t\to\infty}[x-u(t)]\). Denote \(x-u(t)\) by \(v(t)\); then

\[ x=\lim_{t\to\infty} v(t), \tag{27} \]

where \(v(t)\) is the solution of the Cauchy problem

\[ \frac{dv}{dt}=-Av+f,\qquad v(0)=0. \tag{28} \]

In view of (27), (28), as \(R_\varepsilon(A)\) we take the operator

\[ R_\varepsilon(A)f=v\left(\frac{1}{\varepsilon}\right) =\int_0^{+\infty}\frac{1-\exp(-\lambda/\varepsilon)}{\lambda}\,dE_\lambda f, \tag{29} \]

where \(\{E_\lambda\}\) is the resolution of the identity for the operator \(A\), and \(\|R_\varepsilon\|\leq N_\varepsilon=[\varepsilon(1+z_0)]^{-1}\), where \(z_0\) is the positive root of the equation \(1+z_0=\exp z_0\). In this case, from Theorem 2.1 it follows that for \(\omega\in(0,1)\) and any \(f_\delta\), \(\|f-f_\delta\|\leq\delta\), the equality

\[ x=\lim_{\delta\to 0} v\left(\frac{1+z_0}{\delta^\omega}\right), \]

holds, where \(v(\ldots)\) is the solution of the Cauchy problem (28) with \(f\) replaced by \(f_\delta\). For the usefulness of the connection between problem I and the Cauchy problem (28), see the author’s paper [1, 6].

Example 5. Suppose that in Problem II the space \(H\) is \(L_2(-\infty,+\infty)\), \(f=f(t)\), and

\[ Bf=\frac{d^2 f}{dt^2}. \tag{30} \]

Problem II in this case is of interest in geophysics (see [5, a]). If \(\tilde f\) is the Fourier transform and for \(R_\varepsilon(B)\) one takes formula (13), then

\[ R_\varepsilon(B)f=-\frac{1}{2\pi}\int_{|s|<\frac{1}{\varepsilon}} s^2\exp(-ist)\tilde f(s)\,ds. \]

In the present case \(N_\varepsilon=\varepsilon^{-2}\), and Theorem 2.1 (with \(\omega=1/2\)) gives:

\[ \frac{d^2 f}{dt^2} = -\lim_{\delta\to 0}\frac{1}{2\pi} \int_{|s|\le \delta^{-1/4}} s^2\exp(-ist)\tilde f_\delta\,ds, \]

where \(\|f-f_\delta\|\le \delta\).

The case of interest in geophysics, when convergence in the metric of the space of continuous bounded functions is needed, is considered below in § 3.

Example 6. Suppose that in Problem II \(H=L_2(-\infty,+\infty)\), \(f=f(t)\), \(B=-i\,d/dt\). Take in (11)
\(\varphi=i[\exp(-i\varepsilon\lambda)-1]/\varepsilon\lambda\); then, by Parseval’s theorem,
\(\|R_\varepsilon(B)\|\le 1/\varepsilon\), where
\(R_\varepsilon=i[f(t)-f(t+\varepsilon)]/\varepsilon\), and for
\(\varepsilon=\sqrt{\delta}\) Theorem 2.1 finally gives:

\[ \frac{df}{dt} = \lim_{\delta\to 0} \frac{f_\delta(t+\sqrt{\delta})-f_\delta(t)}{\sqrt{\delta}}, \qquad \|f-f_\delta\|_{L_2}\le \delta. \]

Example 7. Suppose that in Problem II the operator \(B\) has the form

\[ B=\sum_{n=1}^{\infty} B_n, \tag{31} \]

where \(B_n\) are unbounded operators. Problem (31) leads to questions of summation of Taylor series, the Cauchy problem for equations of elliptic type, and also to the problem of reducing Problem I to Problem II (see the author’s papers [1, a), б]). If \(R_{\varepsilon_n}(B_n)\) is a regularizer for \(B_n\), then one can prove that

\[ R_\varepsilon(B)= \sum_{1\le n<[1/\varepsilon]} R_{\varepsilon_n}(B_n) \]

(\(\varepsilon_n\) is chosen from the condition
\(\|B_n f-R_{\varepsilon_n}(B_n)f\|\le \varepsilon^2\))
and, by Theorem 2.1, construct a stable method for finding \(Bf\). As an application, consider the case (see [21], p. 46) when

\[ H=L_2(-\infty,+\infty), \qquad f=f(t), \qquad B_n f=a_n f^{(n)}(t), \]

where \(a_n\) are given numbers, \(f\) is a given function, \(f\in L_2\). To construct \(R_\varepsilon(B_n)\) and \(R_\varepsilon(B)\) one may use Example 5, as well as the following result:

\[ R_\varepsilon(B_n)f = \frac{a_n(-1)^n}{2\pi} \int_{-\infty}^{+\infty} s^{2n}\exp(-\varepsilon|s|-its)\tilde f(s)\,ds. \]

§ 3. Problems Ill-Posed on a Linear Manifold

1. Let us consider an important variety of Problems I and II. Let \(x \in L\), where \(L\) is a linear manifold of the original space \(H\), in which Problems I and II were considered. Let a metric \(\rho(x,y)\) be introduced on \(L\),

\[ \rho(x,y)=\rho(x-y,0)\equiv \rho(x-y). \]

For example, if \(H=L_2(-\infty,+\infty)\), then \(L\) is the set of continuous bounded functions, and

\[ \rho(x,y)=\sup_{-\infty<t<\infty}|x(t)-y(t)|. \]

We shall say that a linear operator \(A\) has zero as an accumulation point of its spectrum on \(L\), if there exists a sequence \(\{x_n\}\), \(x_n \in L\), such that \(\rho(x_n)\to\infty\), but \(\|Ax_n\|=1\) as \(n\to\infty\). Analogously, we say that the operator \(B\) is unbounded on the manifold \(L\), if for some sequence \(\{f_n\}\), \(f_n \in H\), we have:

\[ Bf_n \in L,\qquad \|f_n\|=1,\quad \text{but } \rho(Bf_n)\to\infty \text{ as } n\to\infty . \]

At the same time, the case is possible in which the operator \(A\) has a bounded inverse (in the norm of \(H\)), and \(B\) is bounded in \(H\). Then only the case is of interest in which the inequality \(\rho(x,y)\le c\|x-y\|\) is not satisfied with some constant \(c>0\).

Problem \(I_L\). Let the equation

\[ Ax=f \tag{32} \]

be solvable, and moreover uniquely, \(x\in L,\ f\in H\). For any \(\delta\in(0,1)\), indicate such a linear operator \(\widetilde R_\delta(A)\), bounded in \(H\), that when \(\|f-f_\delta\|\le\delta\) one always has \(\widetilde R_\delta(A)f_\delta\in L\) and

\[ \rho(x,\widetilde R_\delta f_\delta)\to 0 \tag{33} \]

as \(\delta\to 0\).

Problem \(II_L\). Let \(f\in D(B)\),

\[ x=Bf, \tag{34} \]

with \(x\in L\). For any \(\delta\in(0,1)\), indicate such a linear operator \(\widetilde R_\delta(B)\), bounded in \(H\), that for all \(f_\delta\), \(\widetilde R_\delta f_\delta\in L\), and when \(\|f-f_\delta\|\le\delta\), \(\delta\to 0\),

\[ \rho(x,\widetilde R_\delta f_\delta)\to 0 . \tag{35} \]

In studying Problems \(I_L\) and \(II_L\) we shall restrict ourselves only to the case where in the space \(H\) we are given the operator \(\Delta\) introduced in § 1.

Theorem 3.1. Let in Problems \(I_L\) and \(II_L\) the operators \(A, B\) and the metric \(\rho(x,y)\) have the following properties:

\(1^\circ\) \(\rho^2(x_1,x_2)\le (\Delta(x_1-x_2),x_1-x_2)\) for \(x_1\in L,\ x_2\in L,\ x_1-x_2\in D(\Delta)\), where the operator \(\Delta\) was introduced in § 1;

\(2^\circ\) the operators \(A\) and \(B\) are self-adjoint in \(H\);

\(3^\circ\) \(x\in D(\Delta)\); in \(H\), respectively, the sets \(D(\Delta)\cap D(A)\) for Problem \(I_L\) and \(D(\Delta)\cap \Delta^{-1}D(B)\) for Problem \(II_L\) are dense everywhere.

Then there exists a Hilbert space \(\widetilde H\) and self-adjoint operators \(\widetilde A\) and \(\widetilde B\) in it such that, from the solvability of certain problems of type I and II, \(\widetilde A x=\widetilde f\) and \(\widetilde B\widetilde f=x\) in \(\widetilde H\), there follows the solvability of Problems \(I_L\) and \(II_L\).

Proof. Consider Problem \(I_L\). Introduce the operator \(\widetilde A=\Delta^{-1}A\) and the space \(\widetilde H\), obtained by completing \(D(\Delta)\) in the scalar product

\((x,y)_1=(\Delta x,y)\). We shall prove that \(\tilde A=\tilde A^*\) in \(\tilde H\) and \(D(\tilde A)=D(A)\cap D(\Delta)\). Since \((\tilde A x,y)_1=(Ax,y)\) for \(x\in D(A)\), \(y\in H\), the functionals \((\tilde A x,y)_1\) and \((Ax,y)\) are bounded for the same \(x\) and \(y\). If \(x\in D(\tilde A)\), \(y\in D(\tilde A)\), then the symmetry condition \((\tilde A x,y)_1=(x,\tilde A y)_1\) is satisfied. Hence \(\tilde A=\tilde A^*\) in \(\tilde H\).

Consider in \(\tilde H\) a problem of type I

\[ \tilde A x=\tilde f, \tag{36} \]

where \(\tilde f=\Delta^{-1}f\). If in the problem \(I_L\), \(\|f-f_\delta\|\leq \delta\), then for \(\tilde f_\delta=\Delta^{-1}f_\delta\) we always have
\[ \|\tilde f-\tilde f_\delta\|_1^2=(f-f_\delta,\Delta^{-1}(f-f_\delta))\leq k\delta^2, \]
where \(k=\|\Delta^{-1}\|\). Solving problem (36) (Theorems 1.1 and 2.1), we construct operators \(\tilde R_\delta(\tilde A)\), \(\|x-\tilde R_\delta(\tilde A)\tilde f_\delta\|\to 0\) as \(\delta\to 0\). By the assumption of the theorem, \(\|z\|_1^2\geq \rho^2(z)\), \(z\in H\); therefore \(\rho(x,\tilde R_\delta(\tilde A)\tilde f_\delta)\to 0\) as \(\delta\to 0\), and the problem \(I_L\) is solved.

For the problem \(II_L\), consider in \(\tilde H\) a problem of type II

\[ x=\tilde B\tilde f, \tag{37} \]

where \(\tilde B=B\Delta\), \(\tilde f=\Delta^{-1}f\). It is not difficult to verify that \((\tilde Bx,y)_1=(x,\tilde B y)_1\) for \(x\in D(\Delta)\) and \(\Delta y\in D(B)\), and then to establish that \(\tilde B=\tilde B^*\), \(D(\tilde B)=\{x;\ x\in D(\Delta),\ \Delta x\in D(B)\}\). Solving problem (37), we also solve \(II_L\).

The theorem is completely proved.

Problems \(I_L\) and \(II_L\) in the case where \(L\) is a set of differentiable functions in domains of \(n\)-dimensional space, for operators \(A\) and \(B\) of a special form, were first studied by A. N. Tikhonov [3, a], and formulations of such problems are found, for example, in [7]. In the case where \(H=\{f(t)\}\), \(f\in L_2(-\infty,+\infty)\), as the operator \(\Delta\) one may take

\[ \Delta f=-f''(t)+f \]

or iterations of this operator.

If \(H=L_2(-a,a)\), \(|a|<\infty\), then \(\Delta f=-f''(t)\) or iterations of the latter operator. The metric in which the approximate solutions then converge is given by embedding theorems, whose usefulness was pointed out in the work of A. B. Bakushinskii [11].

In the case where, in problem \(I_L\), the operator \(A\) is positive, problem (36) can be solved by the method \(x=\lim x_\delta\), \(\delta\to 0\), \(\varepsilon_\delta x_\delta+\tilde A x_\delta=\tilde f_\delta\), i.e. \(\varepsilon_\delta\Delta x_\delta+Ax_\delta=f_\delta\), and for \(\Delta=-d^2/dt^2+E\) we obtain the formula of A. N. Tikhonov [3, a]. We note that D. M. Eidus in [14, a] solved the Helmholtz equation analogously.

For the problem \(II_L\) the following method of solution is practically convenient. The operator \(\tilde B\) in problem (37) is complicated: \(\tilde B=B\Delta\). Replace \(II_L\) by the problem \(\Delta x=\Delta Bf\), \(f\in D(\Delta B)\). In the case where \(B\) and \(\Delta\) are differential operators in a function space, it is usually not difficult to choose such a differential operator \(C\) that \(C^{-1}\) exists, \(\|C^{-1}\|<\infty\), and

\[ \|\Delta B C^{-1}\|<\infty . \tag{38} \]

If \(R_\varepsilon(C)\) is a regularizer of \(C\), then

\[ \rho(Bf,\ BC^{-1}R_\varepsilon(C)f)\to 0 \]

as \(\varepsilon\to 0\), where \(\rho(x_1,x_2)\) defines the metric \(L\). Indeed, it is enough to establish that

\[ \|\Delta Bf-\Delta B C^{-1}R_\varepsilon(C)f\|\to 0 \]

as \(\varepsilon \to 0\). But
\[ \Delta B f-\Delta B C^{-1}R_\varepsilon(C)f = \Delta B C^{-1}\{Cf-R_\varepsilon(C)f\}\to 0 \]
by virtue of (38) and the definition of \(R_\varepsilon(C)\).

1. Let us consider the case, occurring in applications, of problem \(I_L\), when the operator \(A\) commutes with the operator \(\Delta\) from § 1, and the metric of the manifold \(L\) satisfies condition \(1^\circ\) of Theorem 3.1.

We shall show that in this case there is no need to complicate the problem by introducing an auxiliary operator \(\tilde A\), and it is enough merely to “smooth” the approximately known right-hand side \(f_\delta\). This is useful in searching for the simplest methods of solving problem \(I_L\).

Theorem 3.2. Let in problem \(I_L\) we have:

\(1^\circ\) \(\Delta A x=A\Delta x\) for all \(x\in D(\Delta)\cap D(A)\);

\(2^\circ\) there exists a regularizer \(R_\varepsilon(A)\) in \(\bar H\), \(\Delta R_\varepsilon=R_\varepsilon\Delta\);

\(3^\circ\) \(x\in D(\Delta)\).

Let \(R_\omega(\Delta)\) be an arbitrary regularizer of \(\Delta\) in \(H\), \(\tilde R_\omega=\Delta^{-1}\bar R_\omega\), and, finally,
\[ x_{\varepsilon\omega\delta}=R_\varepsilon \tilde R_\omega f_\delta. \]

Then, if \(\|f-f_\delta\|\leqslant \delta\) and with a corresponding choice of \(\varepsilon=\varepsilon(\delta)\) and \(\omega=\omega(\delta)\), we always have, as \(\delta\to 0\),
\[ \rho(x,x_{\varepsilon\omega\delta})\to 0. \tag{39} \]

Proof. From the assumptions of the theorem it follows that \(f\in D(\Delta)\) and
\[ A\Delta x=\Delta f. \]
By the definition of \(R_\varepsilon(A)\) and \(\tilde R_\omega(\Delta)\) we have
\[ x=R_\varepsilon f+o_1(\varepsilon,f); \]
\[ \Delta x=R_\varepsilon\Delta f+o_1(\varepsilon,\Delta f); \]
\[ f_\delta=f+o_2(\delta,f); \]
\[ g=\tilde R_\omega g+o_3(\omega,g) \]
for \(g\in D(\Delta)\), where by \(o_1(\varepsilon,f)\), \(o_2(\delta,f)\), and \(o_3(\omega,g)\) are denoted quantities whose norms in \(H\) tend to \(0\) as \(\varepsilon\to 0\), \(\delta\to 0\), \(\omega\to 0\), respectively, and which depend on \(f\), \(\Delta f\), and \(g\).

To prove (39) it is sufficient to verify that \(\Delta(x-x_{\varepsilon\omega\delta})\to 0\) for some \(\varepsilon(\delta)\), \(\omega(\delta)\), and \(\delta\to 0\). Using the preceding identities and assumption \(2^\circ\) of the theorem, for arbitrary \(\varepsilon,\omega,\delta\in(0,1)\) we have
\[ \Delta(x-x_{\varepsilon\omega\delta}) = o_1(\varepsilon,f)+R_\varepsilon\{\Delta f-\bar R_\omega\Delta f\} - R_\varepsilon\Delta\tilde R_\omega o_2(\delta,f) = \]
\[ = o_1(\varepsilon,f)+R_\varepsilon\{o_3(\omega,\Delta f)-\bar R_\omega o_2(\delta,f)\}. \]

Let \(\|R_\varepsilon\|\leqslant N_\varepsilon\), \(\|\bar R_\omega\|\leqslant M_\omega\), where \(N_\varepsilon\) and \(M_\omega\) are continuous and increase monotonically to \(\infty\) as \(\varepsilon\to 0\). Then (since \(\|o_2(\delta,f)\|\leqslant \delta\))
\[ \|\Delta(x-x_{\varepsilon\omega\delta})\| \leqslant N_\varepsilon\{\delta M_\omega+\|o_3(\omega,\Delta f)\|\} +\|o_1(\varepsilon,\Delta f)\|. \]

For sufficiently small \(\delta\), the equation
\[ M_\omega=\delta^{-1/2} \tag{40} \]
has a root \(\omega_\delta\), and the equation
\[ N_\varepsilon=\{\|o_3(\omega_\delta,\Delta f)\|+\delta^{1/2}\}^{-1/2} \tag{41} \]

has a root \(\varepsilon_\delta\), \(\omega_\delta \to 0\) and \(\varepsilon_\delta \to 0\) as \(\delta \to 0\). Finally, we have

\[ \left\| \Delta (x - x_{\varepsilon_\delta \omega_\delta \delta}) \right\| \le \]

\[ \le \|o_1(\varepsilon_\delta, \Delta f)\| + \left\{ \|o_3(\omega_\delta, \Delta f)\| + \delta^{1/2} \right\}^{1/2}. \tag{42} \]

The theorem is completely proved.

Remarks to Theorem 3.2. \(1^\circ\). Suppose that, in addition to the assumptions of Theorem 3.2, it is known that \(x = Ay\) for some \(y \in H\), \(\|y\| \le c = \mathrm{const}\), and also \(f \in D(\Delta^2)\). Then (see the author’s papers [1, a), b), c)])

\[ \|o_1(\varepsilon, \Delta f)\| \le c\varepsilon, \qquad \|o_3(\omega, \Delta f)\| \le c\omega. \]

Let us take \(R_\varepsilon(A)\) and \(R_\varepsilon(\Delta)\) such that \(N_\varepsilon \le c/\varepsilon\), \(M_\omega \le c/\omega\). Then equations (40), (41), and equations equivalent to them can be solved explicitly, and, taking \(\omega_\delta = \delta^{1/2}\), \(\varepsilon_\delta = \delta^{1/4}\), we obtain the estimate

\[ \rho(x, x_{\varepsilon_\delta \omega_\delta \delta}) \le c\delta^{1/4}, \tag{43} \]

where \(c\) is a certain numerical constant.

\(2^\circ\). The results of Theorem 3.2 and formula (39) also carry over to the problem \(\Pi_L\), if \(x \in D(\Delta)\), \(B\Delta = \Delta B\). For this, in all formulas it suffices to replace \(R_\varepsilon(A)\) by \(R_\varepsilon(B)\). It then turns out that many methods of § 2 give simultaneously convergence in the metric \(L\), if assumption \(3^\circ\) of Theorem 3.2 is satisfied. For example, if \(\|B\Delta^{-1}\| < \infty\), \(f \in D(\Delta)\), then \(R_\varepsilon(B) = B\Delta^{-1}R_\varepsilon(\Delta)\) for any \(R_\varepsilon(\Delta)\). Taking \(R_\varepsilon(\Delta) = \Delta\exp(-\varepsilon\Delta)\), we obtain in the metric \(H\): \(Bf = \lim B\exp(-\varepsilon_\delta\Delta)f_\delta\). But when \(x \in D(\Delta)\), \(B\Delta = \Delta B\), then, taking in Theorem 3.2 \(\overline{R}_\omega(\Delta) = \Delta\exp(-\omega\Delta)\), we obtain \(x_{\varepsilon\omega\delta} = R_{\varepsilon+\omega}(B)f_\delta\), i.e. changing the parameter \(\varepsilon_\delta\) in the old formula yields convergence also in the metric \(L\).

Similarly, if \(\|B\exp(-\Delta^\alpha)\| < \infty\), \(0 < \alpha < 1\), then, on the basis of Theorem 1.2 and Example 3 of § 2, for \(f \in D(\exp(\Delta^\alpha))\)

\[ R_\varepsilon(B) f_\delta \equiv B\exp(-\varepsilon\Delta)f_\delta \to Bf \]

as \(\delta \to 0\) and for a suitable \(\varepsilon = \varepsilon(\delta)\). However, if in addition \(Bf \in D(\Delta)\), \(\Delta B = B\Delta\), then in Theorem 3.2 we take \(R_\varepsilon(B) = B\exp(-\varepsilon\Delta)\), \(\overline{R}_\omega(\Delta) = \Delta\exp(-\omega\Delta)\); for suitable \(\varepsilon\) and \(\omega\), \(x_{\varepsilon\omega\delta} = B\exp(-(\varepsilon+\omega)\Delta)f_\delta \to x\) also in the metric \(L\).

The formula \(B\exp(-\varepsilon\Delta)f_\delta \to Bf\) remains valid also when convergence is required in the metric \(\rho_n(x,y) = (\Delta^n(x-y),(x-y))\), if \(Bf \in D(\Delta^n)\). A similar circumstance for regularizers of the Cauchy problem was noted to some extent by V. K. Ivanov [4, b)]. What was said above about the regularizer \(R_\varepsilon(B) = B\exp(-\varepsilon\Delta)\) carries over also to regularizers \(R_\varepsilon(B)\) of the form (13) and (14).

We give examples illustrating the theorems of the present section.

Example 1. Consider the problem arising in seismometry (see [20]) of finding a function \(x(t)\), \(-\infty < t < +\infty\), from the known function \(F(t)\), where it is known that

\[ \widetilde{F}(\omega) = \widetilde{x}(\omega) U(\omega)\exp(i\gamma(\omega)), \]

where \(U(\omega)\) and \(\gamma(\omega)\) are known continuous functions; \(u_0 < U(\omega) < u_1\), \(u_0\) and \(u_1\) are constants; \(\widetilde{x}\) and \(\widetilde{F}\) are the Fourier transforms of the functions \(x(t)\) and \(F(t)\), \(x \in L_2(-\infty,+\infty)\). In addition, let the sought function \(x \in L\), where \(L\) is the set of continuous bounded functions with metric

\[ \rho(x,y)= \sup_{-\infty < t < \infty} |x(t)-y(t)|, \]

and instead of \(F\) a function \(F_\delta\) is given,

\[ \|F-F_\delta\|_{L_2} \le \delta. \]

It is required, for the given \(F_\delta\), to construct a continuous function \(x_\delta(t)\) such that \(\rho(x,x_\delta)\to 0\) as \(\delta\to 0\).

We apply Theorem 3.2. As \(H\) we take \(L_2\); in addition we require that

\[ x\in D(\Delta), \]

where \(\Delta x=-d^2x(t)/dt^2+x(t)\). In the present case \(\widetilde H\) is \(W_2^1\). Using Parseval’s equality and the Cauchy—Bunyakovsky inequality, we obtain that

\[ (\Delta x,x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}|\widetilde x|^2(1+s^2)\,ds, \]

\[ \|x\|^2\leq 2\pi(\Delta x,x),\qquad |x(t)|\leq 2\pi^{1/2}(\Delta x,x)^{1/2}. \]

We apply Theorem 3.2 to the operator \(BF\):

\[ x=BF=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\widetilde F(\omega)\exp(-it\omega)V(\omega)\,d\omega, \tag{44} \]

where \(V(\omega)=\exp(-iY(\omega))/U(\omega)\).

We use Remark \(2^\circ\) to Theorem 3.2, since \(\|B\|\leq u_0^{-1}\), \(\|B\Delta^{-1}\|<\infty\); then, in the metric \(L\), for a suitable \(\varepsilon=\varepsilon(\delta)\) we have

\[ x=\lim_{\delta\to 0}B\exp(-\varepsilon\Delta)F_\delta(t). \]

Since \(\exp(-\varepsilon\Delta)F_\delta=\exp(-\varepsilon(1+\omega^2))\widetilde F_\delta(\omega)\), it follows that

\[ x_{\varepsilon\delta}=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\exp\bigl(-\varepsilon(1+\omega^2)-it\omega\bigr)V(\omega)\widetilde F_\delta(\omega)\,d\omega. \tag{45} \]

Let us estimate \(|x-x_{\varepsilon\delta}|\) and \(\varepsilon(\delta)\). From (44) and (45), for arbitrary \(\varepsilon\) and \(\delta\),

\[ |x-x_{\varepsilon\delta}|\leq \]

\[ \leq \frac{1}{2\pi}\left\|(1+\omega^2)^{1/2}\widetilde F(\omega)\right\|\,o(1,\varepsilon) +\frac{1}{2\pi}\|F-F_\delta\|\,N_\varepsilon, \tag{46} \]

where \(\|\cdots\|\) is taken in \(L_2(-\infty,+\infty)\);

\[ o^2(1,\varepsilon)=\int_{-\infty}^{+\infty} \frac{\left[1-\exp\bigl(-\varepsilon(1+\omega^2)\bigr)\right]^2}{1+\omega^2}\,d\omega \leq \int \frac{[\cdots]}{1+\omega^2}\,d\omega\leq \]

\[ \text{(with the substitution } \varepsilon\omega^2=\xi^2)\leq \alpha_1\varepsilon^{1/2}, \]

\[ N_\varepsilon^2=\exp(-2\varepsilon)\int \exp(-2\varepsilon\omega^2)\,d\omega\leq \alpha_2\varepsilon^{-1/2}, \]

\(\alpha_1\) and \(\alpha_2\) are numerical constants; \(\|(1+\omega^2)^{1/2}\widetilde F\|^2\leq u_1^2(\Delta x,x)\). Therefore, for arbitrary \(\delta\) and \(\varepsilon\),

\[ |x-x_{\varepsilon\delta}|\leq c\bigl(\varepsilon^{1/4}+\delta\varepsilon^{-1/4}\bigr), \]

where \(c\) is determined mainly by the quantity \((\Delta x,x)\) and can be easily calculated. For \(\varepsilon=\delta^2\), finally,

\[ |x-x_{\delta^2\delta}|<2c\delta^{1/2}. \]

In the problem considered, one can also apply the regularizer (13). The methods considered are applicable in questions of numerical inversion of the Laplace transform (see [10]), as well as in the theory of stable computation of functions from \(L_2\) by the frequency method (see [19, 7, 22]) and in constructing differentiable solutions of the Cauchy problem for the system \(\dot x=Ax+f,\quad x(t)=(x_1(t),\ldots,x_n(t))\), \(A\) being a matrix whose eigenvalues have nonnegative real parts, and \(f\) known with an error in the norm \(L_2(0,+\infty)\), \(t\ge 0\).

Example 2. Consider the problem \(\mathrm{I}_L\), when

\[ Ax \equiv \int_{-\infty}^{+\infty} k(t-s)x(s)\,ds=f(t), \tag{47} \]

\(H=L_2(-\infty,+\infty)\), \(f\in L_2\), the kernel \(k(t)\) is twice differentiable, \(k''(t)\in L_2\), the solution \(x(t)\) is continuous, \(x\in L\). Equations of the form (47) lead to problems on analytic continuation of harmonic functions, the Cauchy problem for the Laplace equation in a strip (as follows from the work of V. K. Ivanov [4, b)]), problems on determining the density of potentials in geophysics [24], and also problems on computing derivatives (see the method of V. N. Strakhov [5, a)]).

Suppose that the Fourier transform \(\tilde k(s)\) of the kernel has the properties

\[ 0<\tilde k<1. \tag{48} \]

Then, using Parseval’s equality, from (48) we find that in equation (47)

\[ 0<A<E, \]

where \(E\) is the identity operator. Solving (47) by a method in which the regularizer is given by the formula

\[ x_{\varepsilon\delta}\equiv R_\varepsilon(A)f_\delta= \sum_{n=0}^{[1/\varepsilon]}(E-A)^n f_\delta \tag{49} \]

(see [2, a), 12, 13, 17, 19, 1, b), c)]), we, as \(\delta\to 0\) and with a suitable choice \(\varepsilon=\varepsilon(\delta)\), obtain the approximation (49) to \(x\) in the mean.

Require that \(x\in D(\Delta)\), \(\Delta x=-x''(t)+x(t)\); then \(x(t)\) is continuous. If now the right-hand side (47) were known exactly, then \(f\in D(\Delta)\), since

\[ \Delta Ax=A\Delta x=\Delta f. \tag{50} \]

It follows from (50) that (49), for \(f_\delta=f\), can be differentiated twice termwise, and (49) gives convergence to \(x\) in the metric \((\Delta(x-y),x-y)\), i.e. also in the metric of continuous bounded functions \(L\), \(x\in L\), as required. If \(f\) is given with error, \(\|f-f_\delta\|_{L_2}\le \delta\), then, by Theorem 3.2, we “smooth” \(f_\delta\) before using formula (49). For simplicity, assuming that \(x=Ay\), \(\|y\|\le 1\), \(\|\Delta^2 f\|\le 1\), by remark 1) to Theorem 3.2 we have

\[ |x-x_\delta|<c\delta^{1/4}, \]

where \(\|f-f_\delta\|\le \delta\);

\[ x_\delta=\sum_{n=0}^{[\delta^{-1/4}]}(E-A)^n \exp(-\sqrt{\delta}\Delta)f_\delta. \]

Example 3. We shall solve problem (47) by the method of reduction to problem \(\mathrm{II}_L\). To this end, applying the Fourier transform and denoting \(G(s)=1/\tilde{k}(s)\), we obtain

\[ x = Bf = \frac{1}{2\pi}\int_{-\infty}^{+\infty} G(s)\exp(-its)\tilde{f}(s)\,ds . \tag{51} \]

Suppose that \(|G(s)| \leq c_1\exp(|s|^\alpha)\), where \(c_1\) and \(\alpha\) are constants, \(0<\alpha<2\), and \(x\in L\). Let us apply to the construction of \(x\) remark \(2^\circ\) to Theorem 3.2. Define the operator \(\exp(\pm\Delta^\alpha)\) by the formula

\[ \exp(\pm\Delta^\alpha)f = \frac{1}{2\pi}\int_{-\infty}^{+\infty} \exp\bigl(\pm(1+s^2)^\alpha - ist\bigr)\tilde{f}(s)\,ds . \]

Since \(\|B\exp(-\Delta^{\alpha_1})\|<\infty\) for \(\alpha<2\alpha_1\), assuming additionally that \(x\in D(\Delta)\) instead of \(x\in L\), we obtain: \(x_\delta\to x\) in the metric of \(L\), where, for some \(\varepsilon=\varepsilon(\delta)\),

\[ x_\delta = \frac{1}{2\pi}\int_{-\infty}^{+\infty} G(s)\exp[-\varepsilon(1+s^2)-ist]\tilde{f}_\delta(s)\,ds . \]

In constructing (51) one can also use the regularizer (13). By similar methods one can compute the moments

\[ \int_{-\infty}^{+\infty} t^n f(t)\,dt, \]

which is equivalent to computing the derivatives of the function \(\tilde{f}(s)\) at \(s=0\). For the application of the latter problem to questions of magnetic prospecting, see [6].

Example 4. Let \(f\in L_2(-\infty,+\infty)\),

\[ x = Bf = f^{(m)}(t) = \varkappa \int_{-\infty}^{+\infty} s^m \exp(-its)\tilde{f}(s)\,ds; \]

where \(\varkappa=(-i)^m/2\pi\); \(\tilde{f}\) is the Fourier transform of the function \(f(t)\); \(x\in L\),

\[ L=\{x;\ \rho^2(x,0)=(\Delta^{2n}x,x)<\infty\}, \qquad \Delta f=-f''(t)+f(t). \]

We illustrate on this example remark \(2^\circ\) to Theorem 3.2, stating that for constructing an approximate \(x\) it is sufficient to take the regularizer (14) for any \(n\).

Let \(x_\delta=R_{\varepsilon_\delta}(B)f_\delta\), where \(\|f-f_\delta\|_{L_2}\leq\delta\); \(\varepsilon_\delta\) will be indicated below,

\[ R_\varepsilon(B)f = \varkappa\int_{|s|<1/\varepsilon} s^m \exp(-its)\tilde{f}(s)\,ds . \]

Theorem 3.2 together with remark \(2^\circ\) guarantees the existence of such an \(\varepsilon_\delta\) that \(\rho^2(x,x_\delta)=\|\Delta^n(x-x_\delta)\|\to0\) as \(\delta\to0\). Since the “analytic structure” of \(R_\varepsilon(B)\) in the present case is known, analogously to the remark to Theorem 2.1 we find the explicit form of \(\varepsilon_\delta\),

\[ \varepsilon_\delta=\delta^{1/2(2n+m)} . \]

If convergence in the weaker metric of the manifold \(L'\), where \(\rho'^2(x,0)=\|\Delta^{n-1}x\|\), is satisfactory, then one can also estimate the rate of approximation of \(x_\delta\) to \(x\). It is not difficult to establish that

\[ \rho'(x,x_\delta)<c\,\delta^{1/2(2n+m)}, \]

where \(c\) is some constant; \(\chi_\delta=R_{\varepsilon_\delta}(B)f_\delta\);

\[ \varepsilon_\delta=\delta^{1/2(2n+m)};\qquad \|f-f_\delta\|_{L_2}\leqslant \delta . \]

The nature of the last phenomenon is that in the present example the solution \(x\) is “representable by sources” through the operator \(B^{-\alpha}\) for some \(\alpha\): \(x=B^{-2n/m}y\) for some \(y\in L_2\) (see [1, a]). We note that convergence in the metric \(L'\) simultaneously gives convergence in the metric \(\rho''\):

\[ \rho''(x,0)=\sum_{p=0}^{n-2}\max_{|t|\leqslant\infty}\left|\frac{d^p x(t)}{dt^p}\right|,\qquad n\geqslant 2 . \]

The estimates given above follow from Parseval’s equality and the following formulas:

\[ \widetilde{\Delta^n R_\varepsilon f}= \begin{cases} (-i)^m(1+s^2)^n s^m \tilde f, & |s|<\dfrac{1}{\varepsilon},\\[6pt] 0, & |s|>\dfrac{1}{\varepsilon}; \end{cases} \]

\[ \widetilde{\Delta^n(B-R_\varepsilon)f}= \begin{cases} (-i)^m(1+s^2)^n s^m \tilde f, & |s|>\dfrac{1}{\varepsilon},\\[6pt] 0, & |s|<\dfrac{1}{\varepsilon}. \end{cases} \]

In conclusion we note that above (Theorem 1.2, Example 3, § 2) the following method for solving the Cauchy problem in a Hilbert space \(H\) was justified. Let, for \(f\in H\), the Cauchy problem

\[ \frac{du}{dt}=Cu,\qquad u=u(t,f),\qquad u(0,f)=f,\qquad t\geqslant 0, \tag{52} \]

be solvable, with \(\|C\Delta^{-\alpha}\|<\infty\), \(0<\alpha<1\), \(f\in D(\exp \Delta^\alpha)\), where \(\Delta\) is defined in § 1. Then

\[ u(t,f)=\lim_{\delta\to 0} v_\delta(t,f_\delta), \]

where

\[ \frac{dv_\delta}{dt}=(-\varepsilon\Delta-C)v_\delta,\qquad v_\delta(0,f_\delta)=f_\delta, \tag{53} \]

\[ \|f-f_\delta\|\leqslant \delta,\qquad \varepsilon=\varepsilon(\delta) \]

depends on \(\delta\) in a suitable way, \(\varepsilon(\delta)\to 0\) as \(\delta\to 0\). In light of the developed theory of problem (52) (see [25, 27]), the (auxiliary) assumption \(f\in D(\exp \Delta^\alpha)\) can apparently be weakened.

As an application, take in (52) and (53)

\[ u=u(t,\xi,\eta),\qquad C=\partial^2/\partial\xi\,\partial\eta \]

(see [26]),

\[ u(0,\xi,\eta)=f(\xi,\eta)\in L_2(E_2),\qquad E_2 \text{ is the plane }(\xi,\eta), \]

\[ \Delta=(\partial^2/\partial\xi^2+\partial^2/\partial\eta^2)^2,\qquad u(t,\xi,\eta)\in L_2(E_2) \]

for every \(t>0\). Then \(\|C\Delta^{-\alpha}\|<\infty\) for every \(\alpha\), \(\alpha\in(1/2,1)\), as follows from the theory of fractional powers of operators [27].

§ 4. Estimates of the Rate of Approximation

As was noted in § 2, the estimate of the rate of convergence in the formula \(R_\varepsilon f_{\delta}\to x\) from Theorem 2.1 is composed of an estimate for \(\|R_\varepsilon\|\) and an estimate for \(\|x-R_\varepsilon f\|\). The quantity \(\|R_\varepsilon\|\) is given by formulas (8) and (12); it may be regarded as known if the analytic form of the function \(\varphi(\varepsilon,\lambda)\) and \(\rho(\varepsilon,\lambda)\) is specified (formulas (15), (21), etc.). The quantity \(\|x-R_\varepsilon f\|\) can be estimated only with additional information about the sought solution \(x\) of problems I and II. In [2, a), b)] the membership of \(x\) in some compact set was used for this purpose.

In the author’s papers [1, a) and b)] a “method of source-representable solutions” was proposed for estimating \(\|x-R_\varepsilon f\|\). For example, it was shown (see [1, c)]) that if \(0<A=A^*<E\), \(x=A^n y\), \(y\in H\), \(n>0\), then

\[ \|x-R_\varepsilon f\|\leq c\|y\|\varepsilon^n, \tag{54} \]

where

\[ f=Ax,\qquad R_\varepsilon=A^{-1}\left(E-(E-A)^{1/\varepsilon}\right),\qquad c<n^n, \]

\[ c=\frac{n^n}{(1+\varepsilon n)^{n+1/\varepsilon}}\sim \left(\frac{n}{e}\right)^n \quad \text{as } \varepsilon\to 0 . \]

The condition \(x=Ay\) singles out, generally speaking, a noncompact set of solutions of the equation \(Ax=f\), even for a continuous operator \(A\) and \(\|y\|\leq 1\). Indeed, let in equation (47) the Fourier transform of the kernel \(\tilde{k}(t)\) have the following property: \(\tilde{k}(s)\geq k_0=\mathrm{const}\) for \(|s|\leq 1\) and \(0<\tilde{k}<1\) for \(|s|\leq\infty\). Choose functions \(y_n(t)\) so that \(\tilde{y}_n(s)=\sqrt n\) for \(|s|\leq 1/n\), but \(\tilde{y}_n=0\) for \(|s|>1/n\). Then \(\|Ay_n\|<1\), but

\[ \|x_n-x_m\|=\|Ay_n-Ay_m\|>k_0\|\tilde{y}_n-\tilde{y}_m\|/\sqrt{2\pi} \]

does not tend to \(0\) as \(n\to\infty\), \(m\to\infty\). In the last example, the functions \(x_n(t)\) for \(\tilde{k}=k_0\) and \(|s|\leq 1\) have the form

\[ x_n(t)=\frac{k_0\sqrt n}{\pi t}\sin\frac{t}{n}, \]

i.e., they oscillate slowly, tending to zero as \(t\to\infty\), \(n\to\infty\).

The physical meaning of the requirement \(x=A^n y\) was clarified for analytic continuation problems in the author’s paper [1, c)].

Let us give one more example. If in the equation \(Ax=f\) the function \(f(t)\) represents the measured temperature of an unbounded rod, and \(x(t)\) is the sought temperature which the rod had one hour before the beginning of the measurements, then \(y(t)\) (under the condition \(x=Ay\)) represents the temperature of the rod two hours before the beginning of the measurements. The condition \(x=Ay\) was also used in [29] and [18].

The condition of source representability of the solution makes it possible to identify regularizers which, in a certain sense, are best in order with respect to the infinitely small quantity \(\|x-R_\varepsilon f\|\) for equal \(\|R_\varepsilon\|\). The theory of optimal methods for solving ill-posed problems has not been sufficiently developed (see [5, a), 21, 30, 32]).

Let us introduce for consideration the class of regularizers \(\mathbf{R}\) for solving problem I. Let \(A=A^*\), \(0<A<E\). For \(R_\varepsilon\in \mathbf{R}\), by formula (7) we have

\[ R_\varepsilon(A)f=\int_0^1 \frac{\rho(\varepsilon,\lambda)}{\lambda}\,dE_\lambda f. \]

Suppose that the function \(\tilde\rho(z,\lambda)=\rho(1/z,\lambda)\) has the following properties:

1) \(\tilde\rho(z,\lambda)\) is measurable and finite with respect to the measures generated by the functions \(\sigma(\lambda)=(E_\lambda v,v)\), \(v\in H\), where \(\{E_\lambda\}\) is the resolution of the identity for the operator \(A\);

2) for all \(z\geqslant 1\) and \(0<\lambda<1\) the conditions

\[ \sup_{\lambda}\left|\frac{\tilde\rho(z,\lambda)}{\lambda}\right|=z,\qquad \inf_{\lambda}\left|\frac{\tilde\rho(z,\lambda)}{\lambda}\right|=\frac{1}{\rho_0}; \]

3) for \(0<\lambda<1\), \(z\geqslant 1\), and \(z\to\infty\), always \(\tilde\rho\to 1\), and for some \(\alpha=\alpha(\tilde\rho)>0\) the inequality

\[ (1-\lambda)^{\alpha z}<|1-\tilde\rho(z,\lambda)|\leqslant 1. \]

For example, the regularizer \(R_\varepsilon=(\varepsilon+A)^{-1}\) belongs to the class \(\mathbf R\). The following is true.

Theorem 4.1. If in problem I the operator \(A\) has the properties \(0<A<E\), \(A=A^*\), and unity is not its eigenvalue, then, when this problem is solved by the methods of §§ 1, 2 using regularizers of the class \(\mathbf R\), the following estimates hold:

\(1^\circ\) if the element \(f\) is known exactly, then

\[ \|x-R_\varepsilon f\|\geqslant \|x\|c^{-\frac1\varepsilon}, \tag{55} \]

where \(c=\|x\|/\|(E-A)^\alpha x\|>1\);

\(2^\circ\) for any \(\delta>0\) and fixed \(f\) there exists an element \(f_\delta\) such that \(\|f-f_\delta\|\leqslant\delta\) and

\[ \|x-R_\varepsilon f_\delta\|\geqslant \|x\|c^{-\frac1\varepsilon}+\frac{\theta\delta}{\varepsilon}, \tag{56} \]

where \(\theta=\theta(A,f)\), \(0<\theta<1\), \(c=\|x\|/\|(E-A)^\alpha x\|\);

\(3^\circ\) for fixed \(R_\varepsilon\in\mathbf R\), for any \(\delta>0\) and arbitrarily small \(\eta>0\) there exist elements \(\hat f\) and \(\hat f_\delta\) such that the equation \(\hat A\hat x=\hat f\) is solvable and

\[ \|\hat x-R_\varepsilon \hat f_\delta\|\geqslant \|\hat x\|c^{-\frac1\varepsilon}+\frac{\delta}{\varepsilon}(1-\eta), \]

where \(c=\|\hat x\|/\|(E-A)^\alpha\hat x\|\);

\(4^\circ\) for any \(n>0\) and all \(f\) from the everywhere dense in \(H\) set

\[ H_n=\{f;\ A^{-(n+1)}f\in H\} \]

(the equation \(Ax=f\) is, obviously, solvable), there exists a regularizer \(R_\varepsilon\in\mathbf R\) such that, when \(\|f-f_\delta\|\leqslant\delta\), the inequality

\[ \|x-R_\varepsilon f_\delta\|\leqslant \tilde c\,\varepsilon^n+\frac{\delta}{\varepsilon}, \tag{57} \]

holds, where \(\tilde c\leqslant \eta^n\|A^{-n}x\|\).

Proof. We shall prove assertion \(1^\circ\). For \(R_\varepsilon\in\mathbf R\) we have

\[ x-R_\varepsilon f=\{E-\tilde\rho(z,A)\}x; \]

\[ \|x-R_\varepsilon f\|^2 = \int_0^1 (1-\tilde\rho(z,\lambda))^2\,d(E_\lambda x,x)\geqslant \]

\[ \geqslant\text{(property 3))}\geqslant \|(E-A)^{\alpha z}x\|^2. \tag{58} \]

Consider the operator \(\widetilde A=(E-A)^\alpha\). The operator \(A\) does not have \(0\) and \(1\) as points of the discrete spectrum; therefore \(Ax\ne x\) and \(\widetilde A x\ne x\) for all \(x\in H\). For \(z>1\) the inequality

\[ \|\widetilde A x\|\leq \|\widetilde A^z x\|^{1/z}\,\|x\|^{1-1/z}. \tag{59} \]

holds.

This inequality is a consequence of Hölder’s inequality with exponents \(z\) and \(z/(z-1)\), applied to the functions \(\lambda^2\omega^{1/z}\) and \(\omega^{1-1/z}\), for the integral

\[ \|\widetilde A x\|^2=\int_0^1 \lambda^2\omega\,d\lambda, \]

where \(\omega=d(\widetilde E_\lambda x,x)/d\lambda\) is a summable function (by assumption); \(\{\widetilde E_\lambda\}\) is the resolution of the identity for the operator \(\widetilde A\). In the general case a limiting passage is necessary. From (58) and (59) we obtain assertion \(1^\circ\).

Let us prove assertion \(2^\circ\). Consider the operator \(R_\varepsilon(A)\):

\[ R_\varepsilon(A)g=\int_0^1 \frac{\widetilde\rho(z,\lambda)}{\lambda}\,dE_\lambda g. \]

Since \(\widetilde\rho\geq \lambda\rho_0\), the inverse operator \(R_\varepsilon^{-1}\) exists, and \(\|R_\varepsilon^{-1}\|\leq \rho_0\). Therefore in the ball \(\|g'\|\leq \delta\), for some \(\beta>0\), the equation

\[ R_\varepsilon g'=\beta(x-R_\varepsilon f) \]

is solvable. Taking \(f_\delta=f-g'\), we then have

\[ \|x-R_\varepsilon f_\delta\|=\|x-R_\varepsilon f\|+\beta\|x-R_\varepsilon f\|. \]

To estimate \(\|x-R_\varepsilon f\|\) we apply inequality (55), and with \(\beta=\delta/\|R_\varepsilon^{-1}x-f\|\) we obtain (56), where

\[ \theta=\frac{\varepsilon\|x-R_\varepsilon f\|} {\|R_\varepsilon^{-1}(x-R_\varepsilon f)\|}\geq \varepsilon/\rho_0, \]

\[ f_\delta=f+\frac{\delta}{\|R_\varepsilon^{-1}x-f\|} \left(-R_\varepsilon^{-1}x+f\right). \]

Let us prove \(3^\circ\). Since \(\|R_\varepsilon\|=1/\varepsilon\), for any \(\delta>0\) and arbitrarily small \(\eta\), \(0<\eta<1\), there exists an element \(h\) such that \(\|h\|<\delta\),

\[ \|R_\varepsilon h\|>\frac{\delta}{\varepsilon}\left(1-\frac{\eta}{2}\right). \tag{60} \]

The operator \(T_\varepsilon=A^{-1}-R_\varepsilon\) has an inverse \(T_\varepsilon^{-1}\), which is defined on those \(g\in D(T_\varepsilon^{-1})\) for which the integral

\[ \|T_\varepsilon^{-1}g\|^2 =\int_0^1 \frac{\lambda^2}{|1-\widetilde\rho(z,\lambda)|^2}\,d(E_\lambda g,g) \]

is bounded.

We prove that \(D(T_\varepsilon^{-1})\) is everywhere dense in \(H\). If \(g=(E-A)^{\alpha z}u\), where \(u\) is an arbitrary element of \(H\), then \(g\in D(T_\varepsilon^{-1})\), since (by virtue of property \(3\))

\[ \|T_\varepsilon^{-1}(E-A)^{\alpha z}u\|^2 =\int_0^1 \frac{\lambda^2(1-\lambda)^{2\alpha z}} {|1-\widetilde\rho(z,\lambda)|^2}\,d(E_\lambda u,u)\leq \]

\[ \leq \int_0^1 \lambda^2\, d(E_\lambda u,u)=\|Au\|^2 . \]

Elements of the form \(g=(E-A)^{\alpha z}u\), \(u\in H\), are everywhere dense in \(H\), since otherwise there exists an element \(v\in H\), \(v\ne 0\), such that \((v,(E-A)^{\alpha z}u)=0\) for every \(u\in H\). But then \((E-A)^{\alpha z}v=0\) and \(v=0\), since \(1\) is not an eigenvalue of the operator \(A\). Since \(D(T_\varepsilon^{-1})\) is everywhere dense in \(H\), there exists an element \(\hat f\) such that \(T_\varepsilon \hat f\) belongs to the \(\delta\eta/4\varepsilon\)-neighborhood of the element \(R_\varepsilon h\),

\[ \|T_\varepsilon \hat f-R_\varepsilon h\|\leq \frac{\delta\eta}{4\varepsilon}. \tag{61} \]

For any \(a\) and \(b\), \(a\in H\), \(b\in H\), \(a\ne 0\), \(b\ne 0\), \(\|a-b\|\leq \|b\|\), the inequality (proved below)

\[ \|a+b\|\geq \|a\|+\|b\|-2\|a-b\| \tag{62} \]

is valid.

In (62) one may put \(a=T_\varepsilon \hat f\), \(b=R_\varepsilon h\), since the condition \(\|a-b\|\leq \|b\|\) is fulfilled by virtue of (61) and (60) for \(0<\eta<1\). Therefore assertion \(3^\circ\) is proved for \(\hat f_\delta=\hat f-h\), \(\hat x=T_\varepsilon \hat f+R_\varepsilon \hat f\).

Let us prove (62). When \(\|a-b\|\leq \|b\|\), one always has \((a-b,b)\leq \|b\|^2\), and therefore \((a,b)<0\), since \((a,b)=\|b\|^2(1+(a-b,b)/\|b\|^2)\). If we take \(b=c+\gamma a\), where \((a,c)=0\), \(\gamma=(a,c)/\|a\|^2\), then

\[ \|c\|\leq \|a-b\|,\quad \gamma>0,\quad \|b\|\leq \|\gamma a\|+\|c\|, \]

\[ \|a+b\|\geq \|a(1+\gamma)\|-\|c\|=\|a\|+\|\gamma a\|-\|c\|, \]

whence (62) follows. Assertion \(3^\circ\) is proved completely.

Let us prove inequality (57). As above, it is not difficult to prove that in \(H\) elements of the form \(f=A^{n+1}v\), \(v\in H\), are everywhere dense. Let \(H_n=\{f;\ A^{-(n+1)}f\in H,\ n>0\}\). For \(f\in H_n\) the equation \(Ax=f\) is solvable and its solution is naturally represented through the operator \(A\), \(x=A^n y\) for some \(y\in H\) and \(n>0\). Take \(R_\varepsilon\in \mathbf R\) of the following form:

\[ R_\varepsilon=A^{-1}\{E-(E-A)^{\frac{1}{\varepsilon}}\}. \tag{63} \]

Here \(\tilde\rho(z,\lambda)=1-(1-\lambda)^z\); conditions 1) and 3) are fulfilled, \(\|R_\varepsilon\|=1/\varepsilon\) and \(\inf(\tilde\rho/\lambda)\geq 1\), since \(\tilde\rho>\lambda\) for \(z\geq 1\). Therefore \(R_\varepsilon\in \mathbf R\). Since

\[ \|x-R_\varepsilon f\|\leq \max_{0\leq \lambda\leq 1}\{\lambda^n(1-\lambda)^{\frac{1}{\varepsilon}}\}\,\|y\|, \]

the estimate (57) is obtained with the following \(\tilde c\):

\[ \tilde c=\frac{n^n}{(1+\varepsilon n)^{n+1/\varepsilon}}. \]

The theorem is proved completely.

Remarks. 1) The condition \(A<E\) in Theorem 4.1 does not reduce generality, since if in problem I the operator \(A\) is unbounded, but self-adjoint and positive, then, considering instead of the problem \(Ax=f\) the problem \(A_1x_1=f\), where \(A_1=A(E+A)^{-1}\), \(x_1=x+f\), we arrive at the case of a bounded \(A_1\).

2) Instead of the regularizer (63), equivalent to the method of successive approximations (see [2, a] and [1, b]), one may take, for example, the regularizer (13)

or (14), which can also be implemented in practice, in particular when summing Fourier series:

\[ R_{\varepsilon}(A)f=\int_{\varepsilon}^{1}\frac{1}{\lambda}\,dE_{\lambda}f. \]

Then, for \(x=A^{n}y,\ n>0,\ y\in H\), we have

\[ \|x-R_{\varepsilon}f_{\delta}\|\leq c\varepsilon^{n}+\delta/\varepsilon, \]

where \(c=\|y\|\).

References

1. Antokhin Yu. T. a) DAN SSSR, 167, No. 4, 727—730, 1966; b) Differential Equations, 2, No. 2, 226—240, 1966; c) Differential Equations, 2, No. 2, 241—251, 1966; d) Differential Equations, 2, No. 4, 525—532, 1966.

2. Lavrent’ev M. M. a) On certain ill-posed problems of mathematical physics, Novosibirsk, 1962; b) Materials for the Soviet-American symposium. Novosibirsk, Siberian Branch of the USSR Academy of Sciences, 1963.

3. Tikhonov A. N. a) DAN SSSR, 151, No. 3, 501—504, 1963; b) DAN SSSR, 156, No. 2, 268—271, 1963; c) DAN SSSR, 162, No. 4, 763—765, 1965; d) DAN SSSR, 153, No. 1, 49—52, 1963.

4. Ivanov V. K. a) Mat. sb., 61 (103): 2, pp. 211—223, 1963; b) Differential Equations, 1, No. 1, 130—136, 1965; c) Materials for the Soviet-American symposium. Novosibirsk, Siberian Branch of the USSR Academy of Sciences, 1963.

5. Strakhov V. N. a) DAN SSSR, 153, No. 3, 533—536, 1963; b) Izv. AN SSSR, ser. geophys., 8, 1206—1221, 1963.

6. Kuznetsov V. V. Geol. and geophys., Siberian Branch of the USSR Academy of Sciences, 8, 136—137, 1965.

7. Gel’fand A. E. Izv. Leningr. Electrotechn. Inst., iss. 53, 158—163, 1964.

8. Fox D., Pucci C. Annal. di Matem. pura ed Appl. (118), ser., 4, 46, 155—182.

9. Yoshiyuki S. IEEE, Trans. Automat. Control, 9, No. 4, 420—426, 1964.

10. Skoblya N. Tables for numerical inversion of the Laplace transform. 1964.

11. Bakushinskii A. B. Journal of Computational Mathematics and Mathematical Physics, 5, No. 4, 744—749, 1965.

12. Bialy H. Z. angew. Math. und Mech., 38, No. 7, 8, 261—263, 1958.

13. Fridman V. M. UMN, vol. XI, iss. 1 (67), 233—234, 1956.

14. Eidus D. M. a) Izv. AN SSSR, ser. matem., 27, No. 5, 1055—1080, 1963; b) Mat. sb., 57 (99), No. 1, 1962, pp. 13—44.

15. Herlitz S. Arkiv. füs., 23, 6, 1963, 571—574.

16. Egorov Yu. V. Journal of Computational Mathematics and Mathematical Physics, 3, No. 5, 887—904, 1963.

17. Gordadze E. G. Tr. Gruz. Polytechn. Inst., 8 (93), 1963, pp. 23—31.

18. Maiorov L. V. Journal of Computational Mathematics and Mathematical Physics, 5, No. 2, 363—365, 1965.

19. Andreev B. A. Izv. AN SSSR, ser. geogr. and geophys., No. 1, 1947.

20. Savarenskii E. F., Fedorov S. A., Gogichaishvili B. V. Izv. AN SSSR, ser. geophys., No. 9, 1340—1347, 1963.

21. Gol’dra G. Ya. Izv. AN SSSR, Physics of the Earth, No. 8, 41—54, 1965.

22. Solodovnikov V. V., Topchiev Yu. I., Krushkova G. V. Frequency method for constructing transient processes. Moscow, Gostekhizdat, 1955.

23. Oskolkov A. P. Vestn. Leningrad State Univ., ser. matem., No. 7, 1963.

24. Novoselitskii V. M. Izv. AN SSSR, Physics of the Earth, No. 5, 25—32, 1965.

25. Sobolevskii P. E. Trudy Moscow Math. Soc., 10, 1961, pp. 297—360.

26. Efimov G. V. ZhETF, 44, No. 6, 2107—2117, 1963.

27. Krein S. G., Sobolevskii P. E. DAN SSSR, 118, No. 2, 233—237, 1958.

28. Pugachev V. S. Proceedings of the All-Union Conference on Probability Theory and Mathematical Statistics. Vilnius, 1962, pp. 233—237.

29. Maslov V. P. Perturbation theory and asymptotic methods. Moscow, 1965.

30. Raspolov O. M. Izv. AN SSSR, Physics of the Earth, 1, 116—120, 1965.

31. Strakhov V. N., Devitsin V. M. Izv. AN SSSR, Physics of the Earth, 4, 60—72, 1965.

32. Dimitrov L. V. Izv. AN SSSR, Physics of the Earth, 9, 89—92, 1965.

33. Garabedyan P. Mechanics, collection of translations. IL, 6 (52), 1958, pp. 23—35.

Received by the editors
October 8, 1966

Moscow