Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

Authors: Babenko, V.¹; Parfinovych, N.V.¹; Pichugov, S.²

Source: Ukrainian Mathematical Journal, Volume 62, Number 3, October 2010 , pp. 343-357(15)

Publisher: Springer

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Abstract:

Let <EquationSource Format="TEX">$$ Cleft( {{mathbb{R}^m}} right) $$</EquationSource> be the space of bounded and continuous functions <EquationSource Format="TEX">$$ x:{mathbb{R}^m} to mathbb{R} $$</EquationSource> equipped with the norm and let e _j, j = 1,…,m, be a standard basis in <EquationSource Format="TEX">$$ {mathbb{R}^m} $$</EquationSource> : Given moduli of continuity ω_j, j = 1,…, m, denote We obtain new sharp Kolmogorov-type inequalities for the norms <EquationSource Format="TEX">$$ left| {D_varepsilon^alpha x} right|C $$</EquationSource> of mixed fractional derivatives of functions <EquationSource Format="TEX">$$ x in cap_{j = 1}^m{H^{j,{omega_j}}} $$</EquationSource> . Some applications of these inequalities are presented.

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s11253-010-0358-y

Affiliations: 1: Dnepropetrovsk National University, Dnepropetrovsk, Ukraine 2: Dnepropetrovsk State Technical University of Railway Transport, Dnepropetrovsk, Ukraine

Publication date: 2010-10-01