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Sharp Estimates of the Constants of Equivalence Between Integral Moduli of Smoothness and K-functionals in the Multivariate Case

Abstract

The equivalence between various types of moduli of smoothness and respective Peetre K-functionals has been actively explored since the 1960s, in view of the importance of this topic for revealing connections among approximation theory, functional analysis and operator theory. The existence of the embedding constants in this equivalence relation (together with one-sided estimates for these constants) has been established in great generality, but the derived one-sided bounds are rather coarse (see, e.g., Johnen and Scherer, in Constructive Theory of Functions of Several Variables. Proc. Conf., Math. Res. Inst. Oberwolfach, 1976, pp. 119–140, Springer, Berlin, 1977 and the references therein). The problem of finding the sharp embedding constants for this equivalence was posed in Dechevsky, C. R. Acad. Bulg. 42(2), 21–24, 1989 and Int. J. Pure Appl. Math. 33(2), 157–186, 2006, where this problem was solved in the particular case of L 2-metric, for real-valued and complex-valued functions of one real variable, with definition domain Ω=ℝ or \(\varOmega =\mathbb{T}\) (the periodic case). In the present paper we extend the results of Dechevsky to the case of several real variables: Ω=ℝn or \(\varOmega =\mathbb{T}^{n}\) , n∈ℕ. We consider two different types of equivalent norms for the Sobolev spaces involved in the K-functional (with and without intermediate mixed partial derivatives) and obtain a separate set of sharp two-sided bounds for the embedding constants in each of these two cases. We also briefly outline how the approach of the present study can be extended to the case of n-dimensional Lie (semi)groups.

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Correspondence to Lubomir T. Dechevsky.

Additional information

The first author is supported in part by the 2007 and 2008 Annual Research Grants of the Priority R&D Group for Mathematical Modelling, Numerical Simulation and Computer Visualization at Narvik University College, Norway.

The second author is supported by a grant from the Norwegian Fellowship Programme for Studies in the High North of SIU—the Norwegian Centre for International Cooperation in Higher Education. This research was part of the work which the second author did at Narvik University College under the supervision of the first author, in relation to the former’s M. Sc. Diploma thesis as a graduate student, jointly at Saint-Petersburg State University, Russia, and Narvik University College, Norway, within a bilateral cooperation programme between the two universities.

Communicated by Vilmos Totik.

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Dechevsky, L.T., Kachkovskiy, I.V. Sharp Estimates of the Constants of Equivalence Between Integral Moduli of Smoothness and K-functionals in the Multivariate Case. Constr Approx 32, 77–90 (2010). https://doi.org/10.1007/s00365-009-9044-4

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Keywords

  • K-functional
  • Modulus of smoothness
  • Embedding
  • Equivalent seminorm
  • Embedding constant
  • Equivalence constant
  • Sharp constant
  • Spectral measure
  • Multivariate
  • Approximation
  • Lagrange multiplier
  • Periodic
  • Laplace operator
  • Shift operator
  • Commutativity
  • Non-commutativity
  • Lebesgue space
  • Sobolev space
  • Hilbert space
  • Lie group
  • Lie algebra
  • Invariant metric
  • Haar measure

Mathematics Subject Classification (2000)

  • 41A44
  • 22E30
  • 41A17
  • 41A65
  • 46B70
  • 46E35
  • 47B15
  • 47B25