Title of article :
Three-monotone spline approximation Original Research Article
Author/Authors :
G.A. Dzyubenko، نويسنده , , K.A. Kopotun، نويسنده , , A.V. Prymak، نويسنده ,
Issue Information :
روزنامه با شماره پیاپی سال 2010
Pages :
16
From page :
2168
To page :
2183
Abstract :
For r≥3r≥3, n∈Nn∈N and each 3-monotone continuous function ff on [a,b][a,b] (i.e., ff is such that its third divided differences [x0,x1,x2,x3]f[x0,x1,x2,x3]f are nonnegative for all choices of distinct points x0,…,x3x0,…,x3 in [a,b][a,b]), we construct a spline ss of degree rr and of minimal defect (i.e., s∈Cr−1[a,b]s∈Cr−1[a,b]) with n−1n−1 equidistant knots in (a,b)(a,b), which is also 3-monotone and satisfies ‖f−s‖L∞[a,b]≤cω4(f,n−1,[a,b])∞,‖f−s‖L∞[a,b]≤cω4(f,n−1,[a,b])∞, Turn MathJax on where ω4(f,t,[a,b])∞ω4(f,t,[a,b])∞ is the (usual) fourth modulus of smoothness of ff in the uniform norm. This answers in the affirmative the question raised in [8, Remark 3], which was the only remaining unproved Jackson-type estimate for uniform 3-monotone approximation by piecewise polynomial functions (ppfs) with uniformly spaced fixed knots. Moreover, we also prove a similar estimate in terms of the Ditzian–Totik fourth modulus of smoothness for splines with Chebyshev knots, and show that these estimates are no longer valid in the case of 3-monotone spline approximation in the LpLp norm with p<∞p<∞. At the same time, positive results in the LpLp case with p<∞p<∞ are still valid if one allows the knots of the approximating ppf to depend on ff while still being controlled. These results confirm that 3-monotone approximation is the transition case between monotone and convex approximation (where most of the results are “positive”) and kk-monotone approximation with k≥4k≥4 (where just about everything is “negative”).
Keywords :
3-monotone approximation by piecewise polynomials and splines , degree of approximation , Jackson-type estimates
Journal title :
Journal of Approximation Theory
Serial Year :
2010
Journal title :
Journal of Approximation Theory
Record number :
852846
Link To Document :
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