The creative work of Andrei N. Kolmogorov is exceptionally wide-ranging. In his studies on trigonometric and orthogonal series, the theory of measure and integral, mathematical logic, approximation theory, geometry, topology, functional analysis, classical mechanics, ergodic theory, superposition of functions, and in formation theory, he solved many conceptual and fundamental problems and posed new questions which gave rise to a great deal of further research. Kolmogorov is one of the founders of the Soviet school of probability theory, mathematical statistics, and the theory of turbulence. In these areas he obtained a number of central results, with many applications to mechanics, geophysics, linguistics and biology, among other subjects. This edition includes Kolmogorov's most important papers on mathematics and the natural sciences. It does not include his philosophical and pedagogical studies, his articles written for the "Bolshaya Sovetskaya Entsiklopediya", his papers on prosody and applications of mathematics or his publications on general questions. The material of this edition was selected and compiled by Kolmogorov himself.

The first volume consists of papers on mathematics and also on turbulence and classical mechanics. The second volume is devoted to probability theory and mathematical statistics. The focus of the third volume is on information theory and the theory of algorithms.

Inhalt

1. A Fourier-Lebesgue series divergent almost everywhere.- 2. On the order of magnitude of the coefficients of Fourier-Lebesgue series.- 3. A remark on the study of the convergence of Fourier series.- 4. On convergence of Fourier series (in collaboration with G.A. Seliverstov).- 5. Axiomatic definition of the integral.- 6. On the limits of generalization of the integral.- 7. On the possibility of a general definition of derivative, integral and summation of divergent series.- 8. On conjugate harmonic functions and Fourier series.- 9. On the tertium non datur principle.- 10. On convergence of Fourier series (in collaboration with G.A. Seliverstov).- 11. A Fourier-Lebesgue series divergent of everywhere.- 12. On convergence of series of orthogonal functions (in collaboration with D.E. Men'shov).- 13. On operations on sets.- 14. On the Denjoy integration process.- 15. On the topological group-theoretic foundation of geometry.- 16. Studies on the concept of integral.- 17. On the notion ofmean.- 18. On the compactness of sets of functions in the case of convergence in mean.- 19. On the interpretation of intuitionistic logic.- 20. On the foundation of projective geometry.- 21. On measure theory.- 22. On points of discontinuity offundions of two variables (in collaboration with LYa. Verchenko).- 23. On normability of a general topological linear space.- 24. Continuation of the study of points of discontinuity of functions of two variables (in collaboration with LYa. Verchenko).- 25. On the convergence of series in orthogonal polynomials.- 26. Laplace transformation in linear spaces.- 27. On the order of magnitude of the remainder term in the Fourier series of differentiable functions.- 28. On the best approximation of functions of a given class.- 29. On duality in combinatorial topology.- 30. Homology rings of complexes and locally bicompact spaces.- 31. Finite coverings of topological spaces (in collaboration with P.S. Aleksandrov).- 32. The Betti groups of locally bicompact spaces.- 33. Properties of the Betti groups of locally bicompact spaces.- 34. The Betti groups of metric spaces.- 35. Relative cycles. The Alexander duality theorem.- 36. On open mappings.- 37. Skew-symmetric forms and topological invariants.- 38. A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem (in collaboration with I.G. Petrovskii and N.S. Piskunov).- 39. A simplified proof of the Birkhoff-Khinchin ergodic theorem.- 40. On inequalities for suprema of consecutive derivatives of an arbitrary function on an infinite interval.- 41. On rings of continuous functions on topological spaces (in collaboration with I.M. Gel'fand).- 42. Curves in a Hilbert space invariant with respect to a one-parameter group of motions.- 43. Wiener spirals and some other interesting curves in a Hilbert space.- 44. Points of local topological character of count ably-multiple open mappings of compacta.- 45. Local structure of turbulencein an incompressible viscous fluid at very large Reynolds numbers.- 46. On the degeneration of isotropic turbulence in an incompressible viscous fluid.- 47. Dissipation of energy in isotropic turbulence.- 48. Equations of turbulent motion in an incompressible fluid.- 49. A remark on Chebyshev polynomials least deviating from a given function.- 50. On the breakage of drops in a turbulent flow.- 51. On dynamical systems with an integral invariant on a torus.- 52. On the preservation of conditionally periodic motions under small variations of the Hamilton function.- 53. The general theory of dynamical systems and classical mechanics.- 54. Some fundamental problems in the approximate and exact representation of functions of one or several variables.- 55. On the representation of continuous functions of several variables as superpositions of continuous functions of a smaller number of variables.- 56. On the representation of continuous functions of several variables as superpositions of continuous functions of one variable and addition.- 57. On the linear dimension of topological vector spaces.- 58. A refinement of the concept of the local structure of turbulence in an incompressible viscous fluid at large Reynolds numbers.- 59. P.S. Aleksandrov and the theory of 8s-operations.- 60. A qualitative study of mathematical models of population dynamics.- Commentary.- List of works by A.N. Kolmogorov.