## Books

Suitable for advanced undergraduates and graduate students in mathematics, this introductory treatment is largely self-contained. Topics include Fourier series, sufficient conditions, the Laplace transform, results of Doetsch and Kober-Erdelyi, Gaussian sums, and Euler's formulas and functional equations. Additional subjects include partial fractions, mock theta functions, Hermite's method, convergence proof, elementary functional relations, multidimensional Poisson summation formula, the modular transformation, and many other areas.

Suitable for advanced undergraduates and graduate students in mathematics, this introductory treatment is largely self-contained. Topics include Fourier series, sufficient conditions, the Laplace transform, results of Doetsch and Kober-Erdelyi, Gaussian sums, and Euler's formulas and functional equations. Additional subjects include partial fractions, mock theta functions, Hermite's method, convergence proof, elementary functional relations, multidimensional Poisson summation formula, the modular transformation, and many other areas.

The linear equation with constant and almost-constant coefficients receives in-depth attention that includes aspects of matrix theory. No previous acquaintance with the theory is necessary, since author Richard Bellman derives the results in matrix theory from the beginning. In regard to the stability of nonlinear systems, results of the linear theory are used to drive the results of Poincaré and Liapounoff. Professor Bellman then surveys important results concerning the boundedness, stability, and asymptotic behavior of second-order linear differential equations. The final chapters explore significant nonlinear differential equations whose solutions may be completely described in terms of asymptotic behavior. Only real solutions of real equations are considered, and the treatment emphasizes the behavior of these solutions as the independent variable increases without limit.

The book opens with J. C. Maxwell's "On Governors" and continues with "The Control of an Elastic Fluid" by H. Bateman; an essay by editors Bellman and Kalaba, "The Work of Lyapunov and Poincaré"; Hurwitz's "On the Conditions Under Which an Equation Has Only Roots With Negative Real Parts"; Nyquist's "Regeneration Theory"; "Feedback — The History of an Idea" by H. W. Bode; a paper on forced oscillations in a circuit by B. van der Pol; "Self-excited Oscillations in Dynamical Systems Possessing Retarded Action" by N. Minorsky; "An Extension of Wiener's Theory of Prediction" by Zadeh and Ragazzini; "Time Optimal Control Systems" by J. P. LaSalle; "On the Theory of Optimal Processes" by Boltyanskii, Gamkrelidze, and Pontryagin; Bellman's "On the Application of the Theory of Dynamic Programming to the Study of Control Processes"; and the editors' study "Dynamic Programming and Adaptive Processes: Mathematical Foundation." Each paper is introduced with a brief account of its significance and with some suggestions for further reading.

methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and

methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.

As a result, the book represents a blend of new methods in general computational analysis,

and specific, but also generic, techniques for study of systems theory ant its particular

branches, such as optimal filtering and information compression.

- Best operator approximation,

- Non-Lagrange interpolation,

- Generic Karhunen-Loeve transform

- Generalised low-rank matrix approximation

- Optimal data compression

- Optimal nonlinear filtering