Constructive Approximation

, Volume 32, Issue 1, pp 77–90 | Cite as

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



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 L2-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.


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 


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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Narvik University CollegeNarvikNorway
  2. 2.Faculty of PhysicsSaint-Petersburg State UniversitySt. PetersburgRussia

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