Fifty years after the original Russian Edition, this classic work
is finally available in English for the general mathematical
audience. This book lays the foundation of what later became “Krein's
Theory of String”. The original ideas stemming from mechanical
considerations are developed with exceptional clarity. A unique feature is that
it can be read profitably by both research mathematicians and engineers.

The authors study in depth small oscillations of one-dimensional continua
with a finite or infinite number of degrees of freedom. They single out an
algebraic property responsible for the qualitative behavior of eigenvalues and
eigenfunctions of one-dimensional continua and introduce a subclass of totally
positive matrices, which they call oscillatory matrices, as well as their
infinite-dimensional generalization and oscillatory kernels. Totally positive
matrices play an important role in several areas of modern mathematics, but
this book is the only source that explains their simple and intuitively
appealing relation to mechanics.

There are two supplements contained in the book, “A Method of
Approximate Calculation of Eigenvalues and Eigenvectors of an Oscillatory
Matrix”, and Krein's famous paper which laid the groundwork for the broad
research area of the inverse spectral problem: “On a Remarkable Problem for a
String with Beads and Continued Fractions of Stieltjes”.

The exposition is self-contained. The first chapter presents all necessary
results (with proofs) on the theory of matrices which are not included in a
standard linear algebra course. The only prerequisite in addition to standard
linear algebra is the theory of linear integral equations used in Chapter 5.
The book is suitable for graduate students, research mathematicians and
engineers interested in ordinary differential equations, integral equations,
and their applications.

Readership

Graduate students, research mathematicians, and engineers
interested in ordinary differential equations, integral equations, and their
applications.