PEOPLE OF SOVIET SCIENCE

ANDREĬ NIKOLAEVICH TIKHONOV

(On the 60th anniversary of his birth)

On October 30, 1966, 60 years will have passed since the birth of the outstanding Soviet mathematician, Hero of Socialist Labor, Lenin Prize laureate, and academician Andreĭ Nikolaevich Tikhonov.

A. N. Tikhonov is the author of fundamental, foundational results in a whole range of areas of contemporary mathematical science.

The first-rate achievements of A. N. Tikhonov in the theory of ordinary differential equations and partial differential equations, in topology and functional analysis, in mathematical problems of geophysics and electrodynamics, in computational mathematics and mathematical physics are universally known. Tikhonov’s scientific work is characterized by a combination of outstanding achievements in the most abstract areas of so-called “pure” mathematics with profound investigations in mathematical disciplines directly connected with the needs of practice.

A. N. Tikhonov was born in the town of Gzhatsk, Smolensk Province. In 1922 he entered the mathematics division of the Faculty of Physics and Mathematics of Moscow University. In 1927, after graduating from the university, he was admitted to graduate study at the Institute of Mathematics of Moscow State University. From 1936 to the present, A. N. Tikhonov has been a professor at Moscow University.

In 1939 A. N. Tikhonov was elected a corresponding member of the Academy of Sciences of the USSR.

In 1953, for outstanding works in mathematical physics, A. N. Tikhonov was awarded by the Government of the USSR the honorary title Hero of Socialist Labor and was awarded the State Prize, First Class.

In April 1966, for outstanding works devoted to the solution of ill-posed problems, A. N. Tikhonov was awarded the Lenin Prize, and in June 1966 he was elected a full member (academician) of the Academy of Sciences of the USSR.

At present A. N. Tikhonov heads two departments at Moscow University: the Department of Mathematics in the Faculty of Physics and the Department of Computational Mathematics in the Faculty of Mechanics and Mathematics.

Let us dwell on the principal scientific achievements of Andreĭ Nikolaevich, doing so in accordance with the scientific orientation of this journal.

we shall cover the works on the theory of differential equations in greater detail, and shall touch upon the remaining results only quite briefly.

The scientific work of Andrei Nikolaevich may be divided into several periods, in each of which his attention was directed to one or another area of contemporary mathematical science.

A. N. Tikhonov devoted the first, earliest period of his scientific work to topology and functional analysis.

At the age of 18–19 A. N. Tikhonov obtained his first topological result, consisting in the proof that every regular topological space with a countable base is normal (and hence metrizable). Approximately two years after this there appeared the results that brought A. N. Tikhonov worldwide renown.

First of all, Tikhonov found a definition of the topological product of an arbitrary set of bicompact spaces. In mathematics the problem of finding the proper definition often plays a decisive role in the whole theory. The “Tikhonov topology,” which underlies the indicated definition, has become firmly established among the fundamental concepts of contemporary mathematics. A. N. Tikhonov placed the definition he had found on a firm foundation by proving the remarkable theorem that the product (in his sense) of any set of bicompact topological spaces is always a bicompact topological space. This remarkable theorem, and a number of subsequent results on the theory of bicompact extensions of topological spaces, undoubtedly make it possible to regard A. N. Tikhonov as one of the outstanding topologists of our time.

A. N. Tikhonov’s topological results play a fundamental role not only for all of modern topology, but also for the theory of topological groups, as well as for functional analysis. These results have very recently found application in a new field of applied mathematics—dynamic programming.

Having no opportunity to dwell in greater detail on the “topological” period of A. N. Tikhonov’s work, we shall turn to a characterization of the second period of his work, connected with the theory of differential equations and with work in the field of geophysics and electrodynamics.

A. N. Tikhonov’s first investigations in the theory of equations in partial derivatives were connected with studies of the heat-conduction equation.

A. N. Tikhonov found, in a certain sense, exhaustive conditions ensuring the uniqueness of the solution of the Cauchy problem for the heat-conduction equation in an infinite domain. It was he who constructed a nontrivial solution \(u(x,t)\) of the indicated equation in the domain \(-\infty < x < \infty\), \(t > 0\), having, as \(|x| \to \infty\), order not exceeding \(Ce^{x^{2+\varepsilon}}\) and equal to zero for \(t = 0\), and proved that the solution of the posed problem is unique if the unknown function \(u(x,t)\) is subject to the condition \(\lim_{|x|\to\infty} u \cdot e^{-Cx^2} = 0\).

In these same works A. N. Tikhonov posed and investigated the inverse problem of heat conduction. He proved a theorem according to which the solution of the heat-conduction equation in the domain \(x > 0\), \(-\infty < t < t_0\) is uniquely determined by the prescribed values \(u(x,t_0)=\varphi(x)\), provided that at least one derivative of the solution with respect to the coordinate is uniformly bounded. The appearance of these investigations of A. N. Tikhonov was stimulated by the study of problems of geophysics devoted to the determination of the geological climate of the earth.

The ideas laid down in A. N. Tikhonov’s investigations devoted to the Cauchy problem for the heat-conduction equation found further devel—

development in the works of a whole series of authors. These include the investigations of Widder, who proved the uniqueness of the solution of the aforementioned Cauchy problem under a semiboundedness condition; O. A. Ladyzhenskaya, who extended A. N. Tikhonov’s results to parabolic equations; S. D. Eidelman, who extended these same results to parabolic systems; and many others.

Very closely connected with the above-mentioned works of A. N. Tikhonov is the joint work of I. M. Gelfand and G. E. Shilov, devoted to properties of Fourier transforms of rapidly increasing functions.

A. N. Tikhonov’s subsequent investigations were devoted to comparing the domains for which the first boundary-value problem for the heat equation and the Dirichlet problem for the Laplace equation and the Helmholtz equation are solvable (in the classical sense).

Defining a fundamental domain as one for which the corresponding problem is solvable, A. N. Tikhonov proved the following propositions:

1) every bounded domain that is fundamental for the heat equation is also a fundamental domain for the Laplace equation;

2) every fundamental domain for the equation $\Delta u-\overline{\lambda}u=0$ for some $\overline{\lambda}>0$ is also a fundamental domain for the equation $\Delta u-\lambda u=0$ for any $\lambda>0$;

3) every domain that is fundamental for the equation $\Delta u-\lambda u=0$ for any $\lambda$ greater than some $\lambda_0$ is fundamental for the heat equation.

In these same works the Green’s function for the heat equation was constructed and many of its properties were studied.

The indicated investigations of A. N. Tikhonov were also continued by other mathematicians. In 1949 O. A. Oleinik and the German mathematician G. Tautz independently proved that a domain fundamental for the Laplace equation is also fundamental for a general elliptic equation of second order with sufficiently smooth coefficients. In 1959 V. A. Il’in proved that a domain fundamental for the Laplace equation is fundamental also for a general self-adjoint hyperbolic and parabolic equation of second order.

A well-known generalization of the indicated works of A. N. Tikhonov was his doctoral dissertation, defended by him in 1936, in which the concept of a functional equation of Volterra type was introduced and the conditions of applicability, for solving this equation, of Picard’s method of successive approximations or the Cauchy–Lipschitz method of polygonal approximations were studied. As applications, a number of heat-conduction problems were considered and, in particular, the problem of the cooling of a body under radiation from its surface according to the Stefan–Boltzmann law. These results were used by V. G. Fesenkov in studying the properties of the surface of the Moon.

To this same and to a somewhat later period belong A. N. Tikhonov’s numerous works devoted to questions of geophysics and electrodynamics.

He constructed a rigorous mathematical theory of the thermoprobe, studied the influence of radioactive decay on the temperature of the Earth’s crust, and solved many problems connected with the development of the theory and methodology of using electromagnetic fields to study the internal structure of the Earth’s crust and, in particular, for prospecting for mineral resources.

In the study of the indicated problems, the problem of stability of inverse problems, considered by A. N. Tikhonov, plays a fundamental role. The principal mathematical method for proving the uniqueness of the solution of the inverse problems considered is the question of the possibility of determining the coefficient \(a(z)\) in the equation \(u''+\lambda a(z)u=0\) \((u(\infty)=0)\) from the given function

\[ f(\lambda)=\frac{u'(0,\lambda)}{u(0,\lambda)}. \]

A. N. Tikhonov solved this question under the condition of piecewise analyticity of the function \(a(z)\) for complex values of the parameter \(\lambda\).

It is interesting to note that in this cycle of A. N. Tikhonov’s works, fundamentally important mathematical results were obtained for the first time concerning the problem of reconstructing a linear differential operator from the properties of its spectrum. These results of A. N. Tikhonov preceded the well-known works of I. M. Gelfand, M. G. Krein, B. M. Levitan, V. A. Marchenko, and other authors.

Approximately the same period includes a cycle of A. N. Tikhonov’s works on electrodynamics, carried out jointly with A. A. Samarskii. These include works on the excitation of electromagnetic oscillations in radio waveguides, in which a general method was proposed for constructing the Green’s function for the system of Maxwell equations in a cylindrical domain with an arbitrary cross-section, and works devoted to the limiting-amplitude principle, according to which, under certain conditions, the unique solution of the Helmholtz equation in an unbounded domain can be determined as the limit as \(t\to\infty\) of the solution of the corresponding Cauchy problem for the wave equation.

These latter investigations were subsequently continued by a number of mathematicians (A. G. Sveshnikov, O. A. Ladyzhenskaya, V. P. Mikhailov, and others).

A. N. Tikhonov was the first to begin studying the question of the behavior of solutions of differential equations with a small parameter at the highest derivative (his first work in this direction was published in 1948). In these works, systems of the form

\[ \frac{dy_i}{dt}=f_i(t,y,z)\quad (i=1,2,\ldots,n);\qquad \mu_j\frac{\partial z_j}{\partial t}=F_j(t,y,z)\quad (j=1,2,\ldots,m), \]

were studied, where \(\mu_j\) are small parameters satisfying the condition \(\mu_{j+1}\ll\mu_j\), and such that as \(\mu\to 0\) there exists the limit of the ratio

\[ \frac{\mu_{j+1}(\mu)}{\mu_j(\mu)}. \]

For such systems, a general formulation of the Cauchy problem was given and criteria were established under which the solutions of the original system tend to the solution of the degenerate system as the parameters tend to zero.

These fundamental investigations of A. N. Tikhonov, first, were widely continued by his students (A. B. Vasil’eva, who constructed an asymptotic expansion of the solution with respect to the small parameter, and V. M. Volosov, who obtained a number of results in the theory of nonlinear oscillations), and, second, stimulated the interest of a whole series of mathematicians in the questions under consideration, among whom one should first of all note the outstanding works of L. S. Pontryagin and his students.

The third period of Andrei Nikolaevich’s scientific work is devoted to computational mathematics. In this field he is an outstanding specialist.

Under the direction of A. N. Tikhonov, numerical methods were developed for solving broad classes of problems of mathematical physics, having—

... of great practical and national-economic importance. Together with A. A. Samarskii, A. N. Tikhonov created a theory of homogeneous difference schemes, which has found important and numerous practical applications.

Let us turn, finally, to the characterization of the brief but very brilliant fourth period of A. N. Tikhonov’s scientific work.

During this time A. N. Tikhonov published a large cycle of works in which he formulated and justified an effective algorithm for solving a broad class of ill-posed problems. In these works A. N. Tikhonov introduced the concept of a class of regularizable ill-posed problems and justified a general method for solving such problems, which he called the regularization method. The regularization method was effectively applied by A. N. Tikhonov to the solution of many fundamental ill-posed problems.

The breadth of Tikhonov’s regularization method is characterized by the following (of course, not claiming completeness) list of problems admitting the application of this method: the problem of finding the solution of an integral and operator equation of the first kind, problems of mathematical economics, inverse problems of potential theory and heat conduction, problems of linear algebra, the problem of analytic continuation of a function, the problem of approximately finding a function and all its derivatives up to some order from perturbed, in \(l_2\), values of its Fourier coefficients, a number of problems from the theory of optimal regulation, problems of geophysical prospecting for minerals, problems of optical and neutron spectroscopy, and many others.

It is also remarkable that the algorithm proposed by A. N. Tikhonov for solving ill-posed problems is extremely well suited for implementation on electronic computers.

In his time P. S. Aleksandrov, whose student A. N. Tikhonov was, said that the brilliant results in topology obtained by Tikhonov at about the age of 20 once again confirm the fact that the most outstanding discoveries in mathematics are often made by very young people.

Now we have grounds to add to this that the brilliant results obtained by A. N. Tikhonov in 1963–1966 also once again confirm the fact that there exist scholars capable of making outstanding mathematical discoveries both at a very young age and at the age of about 60.

Academician Tikhonov is the founder of a large scientific school. Many of his students have already become well-known scholars themselves. The following students of Andrei Nikolaevich are Doctors of Sciences: A. B. Vasil’eva, V. M. Volosov, A. I. Guseinov, V. A. Il’in, E. A. Lyubimova, V. P. Maslov, B. L. Rozhdestvenskii, A. A. Samarskii, and A. G. Sveshnikov. One of Andrei Nikolaevich’s students, A. A. Samarskii, has been elected a Corresponding Member of the Academy of Sciences of the USSR.

Let us wish the jubilarian, who is in the prime of his creative powers, good health and new outstanding scientific results.

V. A. Il’in, A. G. Sveshnikov