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Alexander G. Babenko

Alexander G. Babenko
http://www.imm.uran.ru
Institute of Mathematics and Mechanics of the Ural Branch of Russian Academy of Sciences, Ekaterinburg, Russian Federation

Head of Department of Approximation Theory and Applications, Сhief Researcher

PhD (Phys., Math.), Professor

ScopusID: 57209014890

Scientific interests: Approximation Theory, Extremal Problems for Polynomials and Positive Definite Functions.

Scientific achievements: The (2n+1)-dimensional subspace generated by a 2π-periodic Chebyshev system containing constants and invariant under translation with step 2π/(n+1) was studied. It was proved that the measure of the set of non-negativity points on [0, 2π] of any function in the indicated subspace is more or equal to 2π /(n+1) and this estimate is exact. In the case of a space of trigonometric polynomials of order n, the corresponding problem was posed by L.V. Taikov at the beginning of the 1960s. The exact Jackson – Stechkin inequality in the space L2 of functions on a multidimensional Euclidean sphere and other manifolds is established. Together with V.V. Arestov, the Delsarte problem for positive definite functions was solved. This problem is connected with the problem of the contact number in R4. Together with Yu.V. Kryakin, the problem of integral approximation of the characteristic function of an arbitrary interval by trigonometric polynomials is solved; the analogous problem for the one-sided integral approximation was solved jointly with Yu.V. Kryakin and V.A. Yudin; interesting applications of these results are given.

Main publications:

  1. Babenko A.G. An extremal problem for polynomials // Math. Notes, 1984. Vol. 35, no. 3–4, pp. 181–186. DOI: 10.1007/BF01139914
  2. Babenko A.G. The exact constant in the Jackson inequality in L2 // Math. Notes, 1986. Vol. 39 , no. 5–6, pp. 355–363. DOI: 10.1007/BF01156673
  3. Babenko A.G. Weak-type inequalities for trigonometric polynomials // Trudy Inst. Mat. i Mekh. UrO RAN, 1992. Vol. 2. P. 34–41. [In Russian] (Math. Rev. 95k:42001).
  4. Babenko A.G. Sharp Jackson – Stechkin inequality in L2 for functions on multidimensional spheres // Math. Notes, 1996. Vol. 60, no. 3, pp. 248–263. DOI: 10.1007/BF02320361
  5. Arestov V.V., Babenko A.G. On Delsarte scheme of estimating the contact numbers // Proc. of the Steklov Inst. of Math., 1997. Vol. 219. P. 36–65.
  6. Babenko A.G. Sharp Jackson–Stechkin inequality in L2 on the segment with Jacobi weight and projective spaces // Izvestiya: Mathematics, 1998. Vol. 62, no. 6. P. 1095–1119. DOI: 10.1070/im1998v062n06ABEH000219
  7. Arestov V.V., Babenko A.G. On the optimal point in Jackson's inequality in L2 (-∞,∞) with the second modulus of continuity // East Journal on Approximations, 2004. Vol. 10, no. 1–2, pp. 201–214.
  8. Babenko A.G., Kryakin Yu.V. Integral Approximation of the Characteristic Function of an Interval by Trigonometric Polynomials // Proc. Steklov Inst. Math. (Suppl.), 2009. Vol. 264, suppl. 1. P. S19–S38. DOI: 10.1134/S0081543809050022
  9. Babenko A.G., Kryakin V.Yu., Yudin V.A. One-Sided Approximation in L of the Characteristic Function of an Interval by Trigonometric Polynomials // Proc. Steklov Inst. Math. (Suppl.), 2013. Vol. 280, suppl. 1. P. S39–S52. DOI: 10.1134/S0081543813020041
  10. Babenko A.G., Kryakin V.Yu. and Staszak P.T. Special Moduli of Continuity and the Constant in the Jackson – Stechkin Theorem // Constr. Approx., 2013. Vol. 38, pp. 339–364. DOI: 10.1007/s00365-013-9210-6