Let the Ditzian–Totik moduli of smoothness ω φ r (f, t) p be given as ( 2.1.2) and the equivalent K-functional K r, φ (f, t r ) p be given as ( 2.1.4). As in Section 2.1, we suppose the same definitions for the weight function φ(x) and for the domain D of the operators \(B_{n},S_{n},V _{n},\widehat{B}_{n},\widehat{S}_{n},\widehat{V }_{n}.\) To establish the inverse results for approximation by L n, r we need two Bernstein type inequalities and the Berens–Lorentz lemma , which results we formulate as follows:
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