Lower bounds for the approximation with variation-diminishing splines

Nagler, Johannes; Cerejeiras, Paula; Forster, Brigitte

doi:10.1016/j.jco.2015.08.002

Cited by 8 publications

(9 citation statements)

References 17 publications

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“…We generalize some results of the manuscript [17], where spectral properties of this kind have been shown concretely for the variation-diminishing Schoenberg operator in order to show the limit of the iterates and to prove lower bounds for their approximation error in terms of different moduli of smoothness. If these spectral properties have been established, we apply the famous Theorem of Katznelson and Tzafriri [11] that states that the iterates converge in the operator norm if and only if the spectrum of T has either no points on the unit circle or 1 is the only spectral value on the unit circle.…”

mentioning

confidence: 76%

On the spectrum of positive linear operators with a partition of unity property

Nagler

2015

Journal of Mathematical Analysis and Applications

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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.

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confidence: 76%

On the spectrum of positive linear operators with a partition of unity property

Nagler

2015

Journal of Mathematical Analysis and Applications

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“…Finally, we want to compare our result to a related result that has been shown recently in the case of non‐uniform knots. Nagler et al showed that for all f ∈ C ([0,1]) and

n \to \infty

, the following estimate holds:

ω_{2} (f, \frac{δ_{\min}}{k} \cdot {(\frac{1 - γ_{{normalΔ}_{n}, k}}{d_{k}})}^{1 / 2}) \leq 8 \cdot {‖f - S_{{normalΔ}_{n}, k} f‖}_{[0, 1]},

where

γ_{Δ_{n}, k}

is the second largest eigenvalue of Schoenberg variation‐diminishing spline operator

S_{Δ_{n}, k}

with knot sequence

Δ_{n} = {({}, x_{j})}_{j = - k}^{n - 1}

, ie,

γ_{{normalΔ}_{n}, k} : = \sup \{|λ| : λ \in σ (S_{{normalΔ}_{n}, k}) ∖ \{1\}\},

and

δ_{\min}

denotes the minimal mesh length of the knots,

δ_{\min} : = \min \{(x_{j + 1, k} - x_{j, k}) : j \in \{0, ⋯, n - 1\}\} .

Note that t...…”

Section: The Lower Bound Of the Approximation Error Of The Schoenbergmentioning

confidence: 99%

“…Furthermore, there exists upper bounds for the approximation error with the second‐order modulus of smoothness, see eg, Esser . Recently, such estimates have been shown in Nagler et al and Zapryanova and Tachev, while the results in both articles do not provide computable constants. In this article, we show lower estimates of the uniform spline approximation error in terms of the second‐order modulus of smoothness.…”

Section: Introductionmentioning

confidence: 99%

Detection of C²‐singularities using uniform spline approximations

Nagler

Kähler

2017

Math Methods in App Sciences

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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.

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“…Nagler et al [10] proved the convergence of the iterates of the Shoenberg operator to the operator of linear interpolation at the endpoints of the interval [0,1]. Their approach uses the result of C. Badea [1].…”

Section: Introductionmentioning

confidence: 99%