Approximation by \(B\)-spline convolution operators. A probabilistic approach
type: Item
instance of: scholarly article
title: Approximation by \(B\)-spline convolution operators. A probabilistic approach
zbMATH Open document ID: 1061.41013
author: José A. Adell
author: C. Sangüesa
published in: Journal of Computational and Applied Mathematics
publication date: 2005-02-23T00:00:00Z
review text: The authors are concerned with convolution operators with respect to univariate \(B\)-splines. One of the main features of this paper is that a \(B\)-spline convolution operator is represented as expectation with respect to the linear combination of uniform order statistics. This probabilistic approach has several interesting advantages, i.e.: a) It shows that the \(B\)-spline convolution operator belongs to the general class of Bernstein type operators thus allowing for a unified treatment of their approximation properties. b) It makes easier the computation of moments. c) The proofs are more simple. d) Opens the door to applications of \(B\)-splines in statistics and applied probability. Concerning the contents of the paper, at first the probabilistic representation of univariate \(B\)-splines is recalled. Then, the authors consider \(B\)-spline convolution operators and show a direct inequality in terms of the second modulus of smoothness. For Steklov operators built up from \(B\)-splines with equidistant simple knots strong direct and converse inequalities in the sup-norm are given. Finally, applications to the approximation of functions and to the time ruin probabilities in a classical risk model are presented.
reviewed by: Jaromír Antoch
Mathematics Subject Classification ID: 41A35
Mathematics Subject Classification ID: 62G30
Mathematics Subject Classification ID: 65C60
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 41A15
zbMATH DE Number: 2137443
zbMATH Keywords: \(B\)-spline convolution operator
zbMATH Keywords: Steklov operator
zbMATH Keywords: order statistics
zbMATH Keywords: convergence
zbMATH Keywords: modulus of smoothness
zbMATH Keywords: strong converse inequality
zbMATH Keywords: simultaneous approximation
zbMATH Keywords: divided difference expansion
zbMATH Keywords: ruin probability
zbMATH Keywords: risk model
MaRDI profile type: Publication
full work available at URL: j.cam.2004.04.001
OpenAlex ID: W2155151586
cites work: A strong converse inequality for gamma-type operators
cites work: Upper estimates in direct inequalities for Bernstein-type operators
cites work: LINEAR FUNCTIONS OF UNIFORM ORDER STATISTICS AND B-SPLINES
cites work: Approximation of unbounded functions by a new sequence of linear positive operators
cites work: On the stop-loss and total variation distances between random sums
cites work: Q4273944
cites work: Numerical Finite-Time Ruin Probabilities by the Picard-Lef vre Formula
cites work: Strong converse inequalities
cites work: Q3817079
cites work: Error formulas for divided difference expansions and numerical differentiation
cites work: Nonparametric probability density estimation using normalized b–splines
cites work: Q3768170
cites work: Q3834462
cites work: Two-Sided Bounds for the Finite Time Probability of Ruin
cites work: Q4839745
cites work: Multivariate splines: a probabilistic perspective
cites work: The lower estimate for linear positive operators. I
cites work: Q3672893
cites work: The travel time in carousel systems under the nearest item heuristic
cites work: Equivalence theorem on weighted simultaneous \(L_ p\)-approximation by the method of Kantorovič operators
cites work: The probability of ruin in finite time with discrete claim size distribution
cites work: On the Probability of (Non-) Ruin in Infinite Time
cites work: Approximate distributions of order statistics. With applications to nonparametric statistics
cites work: Lower estimates for centered Bernstein-type operators
cites work: Q3895980
cites work: Convolution of uniform distributions and ruin probability
cites work: Ruin probability by operational calculus
cites work: On convolutions of \(B\)-splines
cites work: Recursive evaluation of aggregate claims distributions.
DOI: 10.1016/J.CAM.2004.04.001
description: scientific article; zbMATH DE number 2137443