Three-Convex Approximation by Free Knot Splines in C [0, 1]

Petrov, Petar

doi:10.1007/s003659900073

Cited by 6 publications

(6 citation statements)

References 3 publications

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“…If it is so, then there is no need in considerations given in § §5-9. This conjecture is true for k = 1, 2 as one can easily check, and our method gives a simpler proof for these cases than in [8] and [13]. For k ≥ 3, the problem is open.…”

Section: Outline Of the Proofmentioning

confidence: 78%

“…Notice that since M k (0, 1) ⊂ C k−2 (0, 1) (see Lemma 3.1), the set S N,r ∩ M k contains functions other than k-monotone polynomials of order r only if r ≥ k. In 1995, Leviatan & Shadrin [8] and Petrov [13], independently, proved that for k = 1, 2 r ≥ k, and 0 < p ≤ ∞, there exists a constant c 0 = O(r) such that, for any f ∈ M k p , k = 1, 2, E…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Naturally, one would expect that the situation is similar for k ≥ 3. However, the technique used in [8], [13] was based on some explicit constructions and some properties of monotone and convex functions which have no straightforward analogues for general k. (Say, for k = 1, 2 the maximum of two k-monotone functions is a k-monotone function, while this is no longer true for larger k.) Petrov [14] has managed to adopt this technique for k = 3 and p = ∞ obtaining an analogue of (1.2), but it became clear that, for general k ∈ N, new ideas are required.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On k-Monotone Approximation by Free Knot Splines

Kopotun¹,

Shadrin²

2003

SIAM J. Math. Anal.

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Let SN,r be the (nonlinear) space of free knot splines of degree r − 1 with at most N pieces in [a, b], and let M k be the class of all k-monotone functions on (a, b), i.e., those functions f for which the kth divided difference [x0,. .. , x k ]f is nonnegative for all choices of (k + 1) distinct points x0,. .. , x k in (a, b). In this paper, we solve the problem of shape preserving approximation of k-monotone functions by splines from SN,r in the Lp-metric, i.e., by splines which are constrained to be k-monotone as well. Namely, we prove that the order of such approximation is essentially the same as that by the non-constrained splines. Precisely, it is shown that, for every k, r, N ∈ N, r ≥ k, and any 0 < p ≤ ∞, there exist constants c0 = c0(r, k) and c1 = c1(r, k, p) such that dist(f, Sc 0 N,r ∩ M k)p ≤ c1 dist (f, SN,r) p ∀f ∈ M k. This extends to all k ∈ N results obtained earlier by Leviatan & Shadrin and by Petrov for k ≤ 3.

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Section: Outline Of the Proofmentioning

confidence: 78%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On k-Monotone Approximation by Free Knot Splines

Kopotun¹,

Shadrin²

2003

SIAM J. Math. Anal.

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“…For 3-monotone approximation of f ∈ ∆ 3 in the L p (quasi-) norm by cubic splines, we can achieve the best possible order of approximation (see (6)), but the location of the knots may depend on f . At the same time, we can still guarantee that the knots are not too close to each other (see (5)), which makes this result different from a constrained free-knot spline approximation (see [7] or [10]). Theorem 1.1.…”

Section: Introduction and Main Resultsmentioning

confidence: 94%

Three-monotone spline approximation

Dzyubenko

Kopotun

2010

Journal of Approximation Theory

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For r ≥ 3, n ∈ N and each 3-monotone continuous function f on [a, b] (i.e., f is such that its third divided differenceswe construct a spline s of degree r and of minimal defect (i.e., s ∈ C r −1 [a, b]) with n − 1 equidistant knots in (a, b), which is also 3-monotone and satisfieswhere ω 4 ( f, t, [a, b]) ∞ is the (usual) fourth modulus of smoothness of f 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 L p norm with p < ∞. At the same time, positive results in the L p case with p < ∞ are still valid if one allows the knots of the approximating ppf to depend on f 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 k-monotone approximation with k ≥ 4 (where just about everything is "negative").

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“…while for s ≥ max{3, r + 1}, we use a construction based on the Gauss quadrature formula (see Petrov [8] for a similar idea). Since x (s−1) is nondecreasing, for each fixed 1 ≤ i ≤ n and s 0 := [(s + 1)/2], there is a quadrature of the form…”

Section: The Upper Boundsmentioning

confidence: 99%

Freeknot splines approximation of Sobolev-type classes of s-monotone functions

Konovalov

Leviatan

2007

Adv Comput Math

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Let I be a finite interval, s ∈ N 0 , and r, ν, n ∈ N. Given a set M, of functions defined on I, denote by s + M the subset of all functions y ∈ M such that the s-difference s τ y(·) is nonnegative on I, ∀τ > 0. Further, denote by W r p the Sobolev class of functions x on I with the seminorm x (r) Lp ≤ 1. Also denote by ν,n , the manifold of all piecewise polynomials of order ν and with n − 1 knots in I. If ν ≥ max{r, s}, 1 ≤ p, q ≤ ∞, and (r, p, q) = (1, 1, ∞), then we give exact orders of the best unconstrained approximation E s + W r p , ν,n Lq and of the best s-monotonicity preserving approximation E s + W r p , s + ν,n Lq .

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Three-Convex Approximation by Free Knot Splines in C [0, 1]

Cited by 6 publications

References 3 publications

On k-Monotone Approximation by Free Knot Splines

On k-Monotone Approximation by Free Knot Splines

Three-monotone spline approximation

Freeknot splines approximation of Sobolev-type classes of s-monotone functions

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