Abstract
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:47A58, 41A17.
Highlights
Introduction and preliminariesThe classical Jackson and Bernstein inequalities express a relation between smoothness modules of functions and properties of their best approximations by polynomials or entire functions of exponential type that can be characterized with the help of Besov norms [, Sections . , . ]
These results are extended to approximations of smooth functions by wavelets, and to approximations of linear operators in Banach spaces by operators with finite ranks [ ], and other similar approximations
The motivation of our work is to extend the Bernstein-Jackson inequalities to cases of best spectral approximations in a Banach space
Summary
Introduction and preliminariesThe classical Jackson and Bernstein inequalities express a relation between smoothness modules of functions and properties of their best approximations by polynomials or entire functions of exponential type that can be characterized with the help of Besov norms [ , Sections . , . ]. Our goal is to investigate a best approximation problem by invariant subspaces of exponential type entire vectors of an arbitrary unbounded closed linear operator A in a Banach space X. Theorem (i) For every p ( ≤ p ≤ ∞) the embedding E∞t (A) ⊂ Epτ (A) with τ > t and the equality E(A) = Ep(A) hold.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
