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Equivalent Norms in Spaces of Functions of Fractional Smoothness on Arbitrary Domains

Authors:	Besov O V

Institution:	(1) V. A. Steklov Mathematics Institute, Russian Academy of Sciences, Russia

Abstract:	In this paper, we study the spaces B _pq ^s (G) and L _pq ^s (G) of functions f with positive exponent of smoothness s > 0 given on a domain $\user1{G} \subset \mathbb{R}^\user1{n}$ . The norms on these spaces are defined via integral norms of the difference of the function f of order m > s treated as a function of the point of the domain and of the difference increment. For an arbitrary domain $\user1{G} \subset \mathbb{R}^\user1{n}$ , we characterize these spaces in terms of the local approximations of the function by polynomials of degree m – 1.

Keywords:	function of fractional smoothness space B _pq ^s space L _pq ^s Banach space integral norm space characterization local approximation of a function by polynomials
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