c Allerton Press, Inc., 2016.
ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2016, Vol. 60, No. 4, pp. 5–9. c I.Yu. Vygodchikova, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 4, pp. 8–13.
Original Russian Text Approximation of a Two-Valued Function
by an Algebraic Polynomial
I. Yu. Vygodchikova1*
1
Saratov State University
ul. Astrakhanskaya 83, Saratov, 410012 Russia
Received August 20, 2014
Abstract—We consider the minimax model of a nonlinear structure for approximating a two-valued
function by an algebraic polynomial. We establish optimality conditions as a strong generalization
of P. L. Chebyshev alternance optimality conditions in approximation of a function by a polynomial.
DOI: 10.3103/S1066369X16040022
Keywords: minimax, nonsmooth analysis, two-valued function, selector, approximating polynomial.
1. The problem. Assume that T = [a, b], a < b, Φ(t) = {y1 (t), y2 (t)}, t ∈ T, are images of a twovalued function Φ(·), y1 (t) and y2 (t) are continuous on T, y2 (t) ≥ y1 (t), t ∈ T, A = (a0 , a1 , . . . , an ) ∈
Rn+1 , pn (A, t) = a0 + a1 t + · · · + an tn (n ≥ 0), and
c(A, t) = |(pn (A, t) − y1 (t))(pn (A, t) − y2 (t))|, t ∈ T.
Let us state the problem
C(A)
= max c(A, t) −→ min .
t∈T
(1)
A∈Rn+1
Putting T = {a = t0 < t1 < · · · < tN = b}, let us state the following auxiliary problem:
C(A) = max c(A, t) −→ min .
(2)
A∈Rn+1
t∈T
The goal of this paper is to establish optimality conditions for problems (1) and (2).
Problem (1) is a generalization of the well-known Chebyshev problem ([1], P. 45)
max |y(t) − pn (A, t)| −→ min
t∈T
A∈Rn+1
and is reducible to it if y(t) = y1 (t) = y2 (t), t ∈ T.
The Chebyshev problem ([1], P. 45) is a particular case of one more problem (e.g., [2, 3]), namely,
ρ(A) = max max{y2 (t) − pn (A, t), pn (A, t) − y1 (t)} −→ min .
t∈T
A∈Rn+1
(3)
Put y1,k = y1 (tk ), y2,k = y2 (tk ), tk ∈ T . Problem (2) is a generalization of the discrete Chebyshev
problem ([1], P. 13) and turns into it with y1,k = y2,k , k ∈ 0, N [4].
The following generalization of the discrete Chebyshev problem is studied sufficiently thoroughly [3]:
ρ(A) = max max{y2,k − pn (A, tk ), pn (A, tk ) − y1,k } −→ min .
A∈Rn+1
k∈0,N
*
E-mail: [email protected].
5
(4)