On 3-monotone approximation by piecewise polynomials |
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Authors: | D Leviatan AV Prymak |
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Institution: | aSchool of Mathematical Sciences, Raymond and Beverley Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel;bDepartment of Mathematical Analysis, Faculty of Mechanics and Mathematics, Kyiv National Taras Shevchenko University, Kyiv 01033, Ukraine |
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Abstract: | We consider 3-monotone approximation by piecewise polynomials with prescribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L1 approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone approximation by piecewise polynomials with prescribed knots. Such estimates are well known for monotone and convex approximation, and to the contrary, they in general are not valid for higher orders of monotonicity. Also we show that any convex piecewise polynomial can be modified to be, in addition, interpolatory, while still preserving the degree of the uniform approximation. Alternatively, we show that we may smooth the approximating piecewise polynomials to be twice continuously differentiable, while still being 3-monotone and still keeping the same degree of approximation. |
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Keywords: | 3-Monotone approximation by piecewise polynomials Degree of approximation |
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