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- Title
Coconvex Approximation of Functions with More than One Inflection Point.
- Authors
Dzyubenko, H. A.; Zalizko, V. D.
- Abstract
Assume that f ∈ C[-1, 1] belongs to C[-1, 1] and changes its convexity at s > 1 different points yi, <MATH>i = \overline{1,s}</MATH>, from (-1, 1). For n ∈ N, n ≥ 2, we construct an algebraic polynomial Pn of order ≤ n that changes its convexity at the same points yi as f and is such that <MATH>\left| f(x) - P_n (x) \right| \leq C(Y)\omega _3 \left( f;\frac{1}{n^2 } + \frac{\sqrt{1 - x^2 }}{n} \right),\quad x \in [ - 1,1],</MATH> where ω3(f; t) is the third modulus of continuity of the function f and C(Y) is a constant that depends only on <MATH>\mathop {\min}\limits_{i = 0, \ldots ,s} \left| y_i - y_{i + 1} \right|</MATH>, y0 = 1, ys + 1 = -1.
- Subjects
CONVEX domains; POLYNOMIALS; MATHEMATICS; ALGEBRA; POINT set theory; CALCULUS of variations
- Publication
Ukrainian Mathematical Journal, 2004, Vol 56, Issue 3, p427
- ISSN
0041-5995
- Publication type
Article
- DOI
10.1023/B:UKMA.0000045688.71949.44