Stechkin-Type Estimate for Nearly Copositive Approximations of Periodic Functions
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STECHKIN-TYPE ESTIMATE FOR NEARLY COPOSITIVE APPROXIMATIONS OF PERIODIC FUNCTIONS G. A. Dzyubenko
UDC 517.5
Assume that a continuous 2⇡ -periodic function f defined on the real axis changes its sign at 2s, s 2 N, points yi : −⇡ y2s < y2s−1 < . . . < y1 < ⇡ and that the other points yi , i 2 Z, are defined by periodicity. Then, for any natural n > N (k, yi ), where N (k, yi ) is a constant that depends only on k 2 N and mini=1,...,2s {yi − yi+1 }, we construct a trigonometric polynomial Pn of order n, which has the same sign as f everywhere, except (possibly) small neighborhoods of the points yi : (yi − ⇡/n, yi + ⇡/n), Pn (yi ) = 0, i 2 Z, and in addition, kf − Pn k c(k, s) !k (f, ⇡/n), where c(k, s) is a constant that depends only on k and s , !k (f, ·) is the k th modulus of smoothness of f, and k · k is the max-norm.
1. Introduction Let C := CR be a space of continuous 2⇡ -periodic functions f : R ! R with uniform norm kf k := kf kR = max f (x) x2R
and let Tn be a space of trigonometric polynomials Pn (x) = a0 +
n X
(aj cos jx + bj sin jx)
j=1
of degree at most n, n 2 N, with aj , bj 2 R. The following statement is known: If a function f belongs to C, then, for any natural n and k, there exists a polynomial Pn 2 Tn such that kf − Pn k c(k) !k (f, ⇡/n) ,
(1.1)
where c(k) is a constant that depends only on k and !k (f, ·) is the modulus of smoothness of order k of the function f. This classical uniform estimation of the approximation of continuous functions by polynomials (without restrictions) was obtained by Jackson (for k = 1), Zygmund (for k = 2), Akhiezer (for k = 2), and Stechkin (for k ≥ 3) (for details, see, e.g., [1, pp. 204–212]). In 1968, Lorentz and Zeller [2] obtained a bell-shaped analog of this estimate for k = 1, i.e., for the approximation of bell-shaped functions (even and nonincreasing on [0, ⇡]) from C by bell-shaped polynomials from Tn . Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine; e-mail: [email protected]. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 5, pp. 628–634, May, 2020. Original article submitted November 14, 2019. 722
0041-5995/20/7205–0722
© 2020
Springer Science+Business Media, LLC
S TECHKIN -T YPE E STIMATE FOR N EARLY C OPOSITIVE A PPROXIMATIONS OF P ERIODIC F UNCTIONS
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This was the origin of the search for other analogs of this estimate with different restrictions imposed on the form of functions and polynomials, such as piecewise positivity, piecewise monotonicity, etc. Assume that 2s, s 2 N, points yi : − ⇡ y2s < y2s−1 < . . . < y1 < ⇡ are fixed on [−⇡, ⇡) and that the other points yi , i 2 Z, are given by the equality yi = yi+2s + 2⇡ (i.e., y0 = y2s + 2⇡, . . . , y2s+1 = y1 − 2⇡, . . .) and Y := Y2s = {yi }i2Z . By ∆(0) (Y ) we denote the set of all f 2 C that are nonnegative on [y1 , y0 ], nonpositive on [y2 , y1 ], nonnegative on [y3 , y2 ], etc. Thus, f 2 ∆(0) (Y ) , f (x)⇧(x) ≥ 0,
x 2 R,
where ⇧(x) := ⇧(x, Y ) :=
2s Y i=1
1 sin (x − yi ) 2
(⇧(x) > 0, x 2 (y1 , y0 ), ⇧ 2 Ts ).
T
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