Besov-type interpolation spaces and appropriate Bernstein-Jackson inequalities, generated by unbounded linear operators in a Banach space, are considered. In the case of the operator of differentiation these spaces and inequalities exactly coincide with the classical ones. Inequalities are applied to a best approximation problem in a Banach space, particularly, to spectral approximations of regular elliptic operators. MSC: Primary 47A58; secondary 41A17
Abstract. We study spaces of analytic functions generated by homogeneous polynomials from the dual space to the symmetric Hilbertian tensor product of a Hilbert space. In particular, we introduce an analogue of the classical Hardy space H 2 on the Hilbert unit ball and investigate spectral decomposition of unitary operators on this space. Also we prove a Wiener-type theorem for an algebra of analytic functions on the Hilbert unit ball.
We establish an improvement of Bernstein-Jackson inequalities by explicitly calculating constants on special approximation scales of analytic vectors of finite exponential types, generated by unbounded operators. Inequalities are applied to analytical estimates of spectral approximations of unbounded operators. Applications to spectral approximations of elliptic and ordinary differential boundary-value problems are shown.
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