Extremal functional interpolation in the mean with least value of thenth derivative for large averaging intervals

Abstract

The smallest numberA<∞ is found such that for any sequenceY={y _k ,k ∈ ℤ} with ¦Δⁿ y _k ¦≤1 there exists au(t), ¦u(t)¦ ≤ A, for which the equationy ⁿ (t)=u(t) (−∞<t<∞) has a solution satisfying the conditions

$$y_k = \frac{1}{h}\int_{ - h/2}^{h/2} {y(k + 1){\mathbf{ }}dt} ,{\mathbf{ }}where{\mathbf{ }}k{\mathbf{ }} \in {\mathbf{ }}\mathbb{Z},{\mathbf{ }}1{\mathbf{ }}< {\mathbf{ }}h{\mathbf{ }}< {\mathbf{ }}2.$$

, wherek ∈ ℤ, 1<h<2.

A similar problem is treated inL _p (−∞, ∞). It is shown that forh=2m (m a natural number) no such finiteA exists.