Sensitivity of best recovery in the Sobolev spaces W^{r,d}_∞, widetilde{W}^{r,d}_∞ for perturbed sampling

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Let I^d be the d–dimensional cube, I^d = [0,1]^d, and let F ∋ f\mapsto Sf∈ L_∞(I^d) be a linear operator acting on the Sobolev space F, where F is either W^{r,d}_∞={f∈ C^{r-1}(I^d): ∥ f∥ _F < ∞} or \widetilde{W}^{r,d}_∞ = {f∈ C^{r-1}({R}^d): f {is 1-periodic w.r.t. each variable, } ∥ f∥ _F < ∞}, where ∥ f∥ _F = Σ_{|m|=r} {esssup}_{x∈ I^d}|\frac{\partial f^{|m|}}{\partial x_1^{m_1}\partial x_2^{m_2}…\partial x_d^{m_d}}(x) |. We assume that the problem elements f satisfy the condition Σ_{ |m|=r}{esssup}_{x∈ I^d} f^{(m)}(x) | ≤ 1 and that S is continuous with respect to the supremum norm.

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