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

Abstract

ABSTRACT. The smallest number A < oo is found such that for any sequence Y = {Yt, k E Z} with IA'tykl < 1 there exists a u(t), lu(t)l < A, for which the equation y'~(t) = u(t) (-oo < t < oo) has a solution satisfying the conditions 1 [ -h/2-y(k+l)dt, where kEZ, 1<h<2. yk = "~ a-~/2A similar problem is treated in Lp(-oo, oo). It is shown that for h = 2m (m a natural number) no such finite A exists.Let 7. be the set of all integers, and let the infinite (in both directions) sequence Y = {yk, k E Z} satisfy the condition II u,,ll,, I,V'u,~l,' _< J.where,Yk+~.Let 0 < h < o0. By ffJI = 9:q(Y, n, p, h) we denote the set of locally Lebesgue integrable functions y(t) satisfying the conditions 1/hi2-~j_h/U(k+t)dt=uk (k e Z). (2)It is required to find the smallest number A = A(n, p, h) such that for any sequence Y satisfying condition (1) there exists a control function =(~), (/_i II=IIL, = I=(t)l p dt < A,such that the equationhas a solution y(t) belonging to the class if/t, i.e:, satisfying conditions (2). For 0 _< h _< 1, this problem was solved by the author in [1, 2]. However, it was stated there in different, though equivalent, terms. In the paper by Sharma and Tsimbalario [3], for h = 0, these results were carried over to nth-order linear differential operators with constant coefficients whose characteristic polynomial only has real roots. In [4, 5] Shevaldin generalized the above-mentioned result of the author to arbitrary nth-order linear differential operators with constant coefficients. For h > 1, the solution of these problems is not known.The foUowing two theorems are proved in this paper.

Approximation by Local Parabolic Splines Constructed on the Basis of Interpolation in the Mean

“…In such case, one uses the interpolation on the average. Aspects of existence, uniqueness, and approximative and extremal properties of such splines were investigated in works [6][7][8]. (Generalizations onto the L-splines see, also, on [9,10]).…”