On Estimates in L2(ℝ) of Mean ν-Widths of Classes of Functions Defined via the Generalized Modulus of Continuity of ωℳ

Abstract

For the classes of functions

\({W^r}\left( {{\omega _{\cal M}},\;{\rm{\Phi }}} \right)\;: = \left\{ {f\; \in \;L_2^r\left(\mathbb{R}\right)\;:\;{\omega _{\cal M}}\left( {{f^{\left( r \right)}},\;t} \right)\; \le \;{\rm{\Phi }}\left( t \right)\;\forall \;t\; \in \;\left( {0,\;\infty } \right)} \right\},\)

where Φ is a majorant and r ∈ ℤ₊, lower and upper bounds for the Bernstein, Kolmogorov, and linear mean ν-widths in the space L₂(ℝ) are obtained. A condition on the majorant Φ under which the exact values of these widths can be calculated is indicated. Several examples illustrating the results are given.