We prove lower bounds for the approximation error of the variationdiminishing Schoenberg operator on the interval [0, 1] in terms of classical moduli of smoothness depending on the degree of the spline basis using a functional analysis based framework. Thereby, we characterize the spectrum of the Schoenberg operator and investigate the asymptotic behavior of its iterates. Finally, we prove the equivalence between the approximation error and the classical second order modulus of smoothness as an improved version of an open conjecture from 2002.
We characterize the spectrum of positive linear operators between Banach function spaces having finite rank and a partition of unity property. Our main result states that all the points in the spectrum are eigenvalues and 1 is the only eigenvalue on the unit circle. Finally, we show that the iterates converge in the uniform operator topology to a projection operator that reproduces constant functions and we provide a simple criterion to obtain the limiting projection operator.We study positive linear operators that have finite rank on a general infinite-dimensional complex Banach function space X that contains the constant function 1 with norm equal to one. In addition, we assume that the associated basis functions of the positive finite-rank operator form a partition of unity. Operators of this kind are used in many applications to approximate functions where only a finite number of samples are available. The partition of unity property guarantees the exact reconstruction of constant functions. Of our interest here is the asymptotic behaviour of iterative applications of the operator and the question whether the limit exists.The asymptotic behaviour of the iterates of positive linear operators has extensively been discussed by many authors. Kelisky and Rivlin [12] have first been considering the limit of iterates of the classical Bernstein operator on the space C ([0, 1]). This result has been extended by Karlin and Ziegler [10] to a more general setting. In [15,16], J. Nagel has examined the asymptotic behaviour of the Bernstein and the Kantorovič operators. Using a contraction principle, Rus [19] has shown an alternative way to prove the convergence of the iterates of the Bernstein operator. The iterates of the Bernstein operators have been also revisited by Badea [2] using spectral properties. Recently, contributions have been made by and by Altomare [1] using methods based on Korovkin-type approximation theory. However, all these results are restricted to the space of continuous functions, i.e., are not applicable for the L p spaces, and there is no general theory that guarantees the existence of the limit of the iterates.
MOS Classification: 41A15; 41A27; 41A36; 41A40For the detection of C 2 -singularities, we present lower estimates for the error in Schoenberg variation-diminishing spline approximation with equidistant knots in terms of the classical second-order modulus of smoothness. To this end, we investigate the behaviour of the iterates of the Schoenberg operator. In addition, we show an upper bound of the second-order derivative of these iterative approximations. Finally, we provide an example of how to detect singularities based on the decay rate of the approximation error.
We consider discrete planar curves as they appear in segmented images. In the literature, the curvature of such curves is often estimated via B-spline approximations or by interpolation schemes, while to the best of our knowledge current methods lack of a proof of convergence, see [2,3]. We will not only proof the convergence of our method in the uniform norm for smooth curves, we will also show that our method is able to detect critical points (C 2 -singularities) of our given discrete data, i.e., points where the curvature is undefined. The main idea is to approximate the curve such that the shape of the curve is preserved. Here, we use the Schoenberg splines because of the freedom to choose the knots arbitrarily and because of their variation-diminishing property that leads to an approximation which preserves positivity, monotonicity and convexity. Shape preserving approximation with Schoenberg splinesLet n, k > 0 be integers, let the knot sequence ∆ n = {t j } n+k j=−k such that 0 = t −k = · · · = t 0 < t 1 < · · · < t n = · · · = t n+k = 1 and consider for j = −k, . . . , n − 1 the knot averages ξ j,k := k l=1 t j+l /k and the B-splines N j,k (t) which are normalized such that they form a partition of unity. Then the Schoenberg operator of degree k is defined forNote that the operator can be similarly defined for vector valued functions on [0, 1], as will be needed in Section 2. It has been shown by Marsden [4] that this operator preserves the positivity, the monotonicity, and the convexity of continuous functions. Furthermore, Marsden has shown that not only the approximation converges uniformly on [0, 1] but also the first two derivatives, while in the latter case the convergence holds only on compact subsets of (0, 1). For more details and properties of the Schoenberg operator we refer to [4,6], shape preserving approximations are discussed in [1]. Curvature approximationIn the following we will show that the curvature derived by the spline approximation (1) converges uniformly towards the real curvature on all compact subsets of (0, 1). To the best of our knowledge existing methods do not provide a proof of convergence, while a comparative study based on numerical experiments can be found in [2]. For a more comprehensive overview of existing methods we refer to [3] and the references therein.Let us consider the planar curve α : [0, 1] → R 2 , α(t) = (x(t), y(t)) T , such that α is twice continuously differentiable. Then the signed curvature is defined byWe approximate the signed curvature of α by the curvature of its spline approximation S ∆n,k α κ ∆n,k (t) = det(DS ∆n,k α(t), D 2 S ∆n,k α(t)) DS ∆n,k α(t) 3 .In the following we will show that |κ(t) − κ ∆n,k (t)| → 0 uniformly on all compact subsets of (0, 1) for n → ∞. To this end, let us denote by δ min := min(t j+1 − t j ) and by δ max := max(t j+1 − t j ), where j = 0, . . . , n − 1, the minimal and the maximal distance of two consecutive knots of the sequence ∆ n .Lemma 2.1 Suppose f ∈ C 2 ([0, 1]) and k > 2. Then the derivatives of the spline appr...
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