A new technique is developed for estimating functionals by moduli of continuity. The generalized Jackson inequality A σ−0 (f) ≤ 1 2m m m−1 k=0 K 2k (γπ) 2k ν k m + K 2m (γπ) 2m ν m m 2 2m ω 2m f, γπ σ is an example of such an estimate. Here r, m ∈ N, σ, γ > 0, a function f is uniformly continuous and bounded on R, A σ−0 is the best uniform approximation by entire functions of type less than σ, ω 2m is a uniform modulus of continuity of order 2m, K s are the Favard constants, and ν m = 8 2m m (m−1)/2 l=0 2m m−2l−1 (2l + 1) 2 , where x is the entire part of x. Similar inequalities are obtained for best approximations of periodic functions by splines. In some cases, the constants in inequalities are close to optimal.
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