Abstract
For every 3-convex piecewise-polynomial function s of degree ≤4 with n equidistant knots on [0, 1] we construct a 3-convex spline s 1 (s 1 ∈ C (3)) of degree ≤4 with the same knots that satisfies the inequality
where c is an absolute constant and ω5 is the modulus of smoothness of the fifth order.
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REFERENCES
A. W. Roberts and D. E. Varbeg, Convex Functions, Academic Press, New York (1973).
R. A. DeVore, “Monotone approximation by splines,” SIAM J. Math. Anal., 8, No.5, 891–905 (1977).
R. K. Beatson, “Convex approximation by splines,” SIAM J. Math. Anal., 12, 549–559 (1981).
Y. K. Hu, “Convex approximation by quadratic splines,” J. Approxim. Theor., 74, 69–82 (1993).
K. A. Kopotun, “Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials,” Contstr. Approxim., 10, 153–178 (1994).
I. A. Shevchuk, “One construction of cubic convex spline,” Proc. ICAOR, 1, 357–368 (1997).
V. N. Konovalov and D. Leviatan, “Shape-preserving widths of Sobolev-type classes of s-monotone functions on a finite interval,” Isr. J. Math., 133, 239–268 (2003).
V. N. Konovalov and D. Leviatan, “Estimates on the approximation of 3-monotone function by 3-monotone quadratic splines,” East J. Approxim., 7, 333–349 (2001).
A. V. Prymak, “Three-convex approximation by quadratic splines with arbitrary fixed knots,” East J. Approxim., 8, No.2, 185–196 (2002).
D. Leviatan and A. V. Prymak, “On 3-monotone approximation by piecewise polynomials,” J. Approxim. Theor., 133, 97–121 (2005).
A. V. Bondarenko, “Jackson type inequality in 3-convex approximation,” East J. Approxim., 8, No.3, 291–302 (2002).
J. Gilewicz, Yu. V. Kryakin, and I. A. Shevchuk, “Boundedness by 3 of the Whitney interpolation constant,” J. Approxim. Theor., 119, 271–290 (2002).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 277–283, February, 2005.
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Prymak, A.V. Shape-Preserving Smoothing of 3-Convex Splines of Degree 4. Ukr Math J 57, 331–339 (2005). https://doi.org/10.1007/s11253-005-0193-8
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DOI: https://doi.org/10.1007/s11253-005-0193-8