Speciality:
01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date:
7.03.1933
Website: https://www.math.spbu.ru/user/analysis/pers/havin.html Keywords: spaces of analytic functions; Cauchy type integrals; separation of singularities of analytic functions; Taylor coefficients of various classes of analytic functions; Hardy classes; inner-outer factorization of analytic functions; approximation by rational, analytic and harmonic functions; analytic capacity; non linear potential theory; the Cauchy problem for the Laplace operator; harmonic fields and differential forms; uncertainty principle in Fourier analysis; coinvariant (model) subspaces of the Hardy space.
Subject:
A solution of Golubev"s problem was given on Laurent generalized representation of an arbitrary function analytic in the complement of a simple rectifiable arc. A new approach to separation of singularities of analytic functions (the Poincare–Aronszajn theorem) was found. Conditions were found ensuring the separation of singularities preserving the uniform boundedness (joint work with A. Nersessyan). For some classes of functions analytic in a disc the invariance under division by an inner function is proved. The phenomenon of the twofold decrease of smoothness of an analytic function compared with the smoothness of its modulus on the boundary is studied ; "the smoothness" can be understood in various senses. Integral analogs of the Vitushkin theorem on the uniform rational approximation were obtained; in the mean square case the role of analytic capacity is played by the classical logarithmic capacity. In a series of joint publications with V. G. Maz'ya "the nonlinear potrential theory" was founded and applied to problems of uniqueness and approximation for analytic and harmonic functions. Some problems posed by S. N. Mergelyan were solved on the solutions of the Cauchy problem for the Laplace equation. (Some rather complete results on "the free solvability" of the Cauchy problem for harmonic functions of two varables were obtained in a joint work with J. Bourgain, A. Aleksandrov, M. Giesecke, and Yu. Vymenets). In a joint work with B. Joericke free interpolation by harmonic functions (in the spirit of Carleson–Garnett) was studied when the interpolation data are defined on a subset of the closed domain and on a part of the boudary. In a series of joint works with B. Joericke some "uncertainty principles" were proved for the convolution integral operators. A monograph (by V. Havin and B. Joericke) is devoted to the uncertainty principle in Fourier analysis ("a non zero function and its Fourier image cannot be too small simultaneously"). In a series of joint works with A. Presa, Ye. Malinnikova, and S. Smirnov approximation properties of harmonic vector fields and differential forms were studied; multidimensional analogs of the Runge theorem and Hartogs–Rosental theorem were proved, and it was shown that the analog of Bishop's localization principle is not valid in dimensions higher than two. In a joint work with J. Mashreghi some "multiplier theorems" (in the spirit of Beurling–Malliavin) for "model subspaces" (i.e. inverse shift invariant) of the Hardy space were obtained. An uncertainty principle for M. Riesz potentials on the line was found; its sharpness was proved in a joint work with D. Belyaev.
Main publications:
Havin V. Golubev series and analyticity on a continuum // Lecture Notes in Math., 1984, 1043, 670–673.
Havin V., Joericke B. The Uncertainty Principle in Harmonic Analysis. Springer-Verlag, 1994.
Khavin V. P., Smirnov S. K. Approximation and extension problems for some classes of vector fields // St. Petersburg Math. J., 1999, 10(3), 507–528.
Beliaev D. B., Havin V. P. On the uncertainty principle for M. Riesz potentials // Arkiv for Mat., 2001, 39(2), 223–243.
