Inequalities for the Derivatives * †
نویسنده
چکیده
The following question is studied and answered: Is it possible to stably approximate f if one knows: 1) f δ ∈ L ∞ (R) such that f − f δ < δ, and 2) f ∈ C ∞ (R), f + f ≤ c? Here f := sup x∈R |f (x)| and c > 0 is a given constant. By a stable approximation one means L δ f δ − f ≤ η(δ) → 0 as δ → 0. By L δ f δ one denotes an estimate of f. The basic result of this paper is the inequality for L δ f δ − f , a proof of the impossibility to approximate stably f given the above data 1) and 2), and a derivation of the inequality η(δ) ≤ cδ a 1+a if 2) is replaced by f 1+a ≤ m 1+a , 0 < a ≤ 1. An explicit formula for the estimate L δ f δ is given.
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