Shape-preserving Approximation Research Articles

On Conic Equations Under Bernstein Operators

One of the most important problems in approximation theory in mathematical analysis is the determination of sequences of polynomials that converge to functions and have the same geometric properties. This type of approximation is called the shape-preserving approximation. These types of problems are usually handled depending on the convexity of the functions, the degree of smoothness depending on the order of differentiability, or whether it satisfies a functional equation. The problem addressed in this paper belongs to the third class. A quadratic bivariate algebraic equation denotes geometrically some well-known shapes such as circles, ellipses, hyperbolas and parabolas. Such equations are known as conic equations. In this study, it is investigated whether conic equations transform into a conic equation under bivariate Bernstein polynomials, and if so, which conic equation it transforms into.

Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update

It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same.

Characterizing best isotone approximations in [formula omitted] spaces, [formula omitted

We characterize best approximations in Lp(1≤p<∞) from a class of convex subcones of the convex cone of nonnegative functions and the convex cone of increasing functions; in particular, from the convex cones themselves. These subcones arise naturally in applications of constrained or shape-preserving approximation.

On saturation effect for linear shape-preserving approximation in Sobolev spaces

On saturation effect for linear shape-preserving approximation in Sobolev spaces

Iterated Bernstein operators for distribution function and density estimation: Balancing between the number of iterations and the polynomial degree

Despite its slow convergence, the use of the Bernstein polynomial approximation is becoming more frequent in Statistics, especially for density estimation of compactly supported probability distributions. This is due to its numerous attractive properties, from both an approximation (uniform shape-preserving approximation, etc.) and a statistical (bona fide estimation, low boundary bias, etc.) point of view. An original method for estimating distribution functions and densities with Bernstein polynomials is proposed, which takes advantage of results about the eigenstructure of the Bernstein operator to refine a convergence acceleration method. Furthermore, an original data-driven method for choosing the degree of the polynomial is worked out. The method is successfully applied to two data-sets which are important benchmarks in the field of Density Estimation.

On the degree of piecewise shape-preserving approximation by polynomials

We show that the degree of piecewise q-coconvex approximation En(q)(f,Ys) of a q-convex function by algebraic polynomials of degree <n asymptotically depends only on α,Ys and q, nαEn(q)(f,Ys)≤c(α,s,q),n≥N∗(α,Ys,q), assuming this inequality holds for an N∗ dependent on f and that the degree of unconstrained approximation has the same decrease rate:nαEn(f)≤1,n≥N with N≤s+q+1. In particular, cases α>1,N=s+2,q=1, and α>2,N=s+3,q=2 complement earlier results on comonotone and coconvex approximation by polynomials respectively.

A new approach for approximate implicitization of parametric curves

In this paper, we present a new approach to solve approximate implicitization of parametric curves. The basic idea is to divide the normal parametric curve into several curve segments at three types of critical points and then use multiquadric quasi-interpolation to approximate each curve segment. Meanwhile, we interpolate two endpoints of each segment using compactly supported radial basis functions in order to maintain the continuity of the adjacent curve segments. The resulting implicit curves possess certain shape preserving and good approximation behaviors.

Shape-preserving dynamic programming

Dynamic programming is the essential tool in dynamic economic analysis. Problems such as portfolio allocation for individuals and optimal growth of national economies are typical examples. Numerical methods typically approximate the value function and use value function iteration to compute the value function for the optimal policy. Polynomial approximations are natural choices for approximating value functions when we know that the true value function is smooth. However, numerical value function iteration with polynomial approximations is unstable because standard methods such as interpolation and least squares fitting do not preserve shape. We introduce shape-preserving approximation methods that stabilize value function iteration, and are generally faster than previous stable methods such as piecewise linear interpolation.

A shape-preserving approximation by weighted cubic splines

This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data. Such spline interpolations are a useful and efficient tool in computer-aided design when control of tension on intervals connecting interpolation points is needed. The error bounds for interpolating weighted splines are obtained. A method for automatic selection of the weights is presented that permits preservation of the monotonicity and convexity of the data. The weighted B-spline basis is also well suited for generation of freeform curves, in the same way as the usual B-splines. By using recurrence relations we derive weighted B-splines and give a three-point local approximation formula that is exact for first-degree polynomials. The resulting curves satisfy the convex hull property, they are piecewise cubics, and the curves can be locally controlled with interval tension in a computationally efficient manner.

On shape-preserving properties and simultaneous approximation of Stancu operator

In the present paper, the shape-preserving properties and the monotonicity for convex functions of Stancu operator are given. Moreover, the simultaneous approximation problems of this operator are also considered.

Refining cubic parametric B-splines

We present refinement properties of cubic parametric B-splines: a family of functions introduced some years ago in the context of shape-preserving approximation of curves. The refinement equations determine a related subdivision scheme which turns out to be irregular, nonstationary and nonuniform.

Shape-Preserving Approximation of Spatial Data

We present a new method for the construction of shape-preserving curves approximating a given set of 3D data, based on the space of “quintic like” polynomial splines with variable degrees recently introduced in [7]. These splines – which are C3 and therefore curvature and torsion continuous – possess a very simple geometric structure, which permits to easily handle the shape-constraints.

Shape-preserving approximation of multiscale univariate data by cubic L1 spline fits

Spline fits are calculated by minimizing a data fitting functional over a manifold of splines. Smoothing splines are calculated by minimizing a linear combination of a data fitting functional and an interpolating spline functional. For multiscale data, that is, data with abrupt changes in magnitude and/or spacing, currently available spline fits and smoothing splines typically have extraneous oscillation. In this paper, we introduce a class of cubic L 1 spline fits that do not have extraneous oscillation. In contrast to conventional cubic spline fits, which are based on L 2 functionals (that is, on quadratic functionals, which consist of sums and integrals of squares), cubic L 1 splines are based on minimizing L 1 functionals (sums and integrals of absolute values). Cubic L 1 spline fits are calculated by a Lagrange-multiplier-based primal affine (interior point) algorithm. Cubic L 1 spline fits are compared with conventional L 2 cubic spline fits and with cubic L 1 and L 2 smoothing splines. Computational results indicate that L 1 spline fits preserve shape well, nearly as well as L 1 smoothing splines with good balance parameters and better than conventional L 2 spline fits and L 2 smoothing splines. L 1 spline fits are computationally more expensive than L 1 smoothing splines and considerably more expensive than L 2 spline fits and L 2 smoothing splines. However, L 1 spline fits require only half the storage required by smoothing splines and do not involve a balance parameter that is required by smoothing splines.

Interpolation of Cones and Shape-Preserving Approximation

In this paper, we present an approach to shape-preserving approximation based on interpolation space theory. In particular, we prove the corresponding approximation result related to the intersection property of the cone of nonnegative functions with respect to the couple ( L p , B α,∞ p ).

Shape-Preserving Approximation by Space Curves

This paper describes a new method for the construction of C2 shape-preserving curves which approximate an ordered set of data in R3. The curves are obtained using the variable degree polynomial spline spaces recently described in [5].

Shape-preserving approximation by polynomials

We are going to survey recent developments and achievements in shape-preserving approximation by polynomials. We wish to approximate a function f defined on a finite interval, say [−1,1], while preserving certain intrinsic “shape” properties. To be specific we demand that the approximation process preserves properties of f, like its sign in all or part of the interval, its monotonicity, convexity, etc. We will refer to these properties as the shape of the function.

Shape-preserving approximation inL p

Shape-preserving approximation inL p

Convexity Preserving Approximation by Free Knot Splines

In this paper the order of shape-preserving approximation of functions f in Sobolev space by free knot splines is studied. The main result is that the k-convexity of f for general k can be preserved, and the optimal order of approximation $n^{ - r} $ can be retained at the same time.

Shape-preserving quasi-interpolation and interpolation by box spline surfaces

This paper is devoted to the study of shape-preserving approximation and interpolation of functions by box spline surfaces on three and four directional meshes. The properties of positivity, monotonicity, and convexity are considered. A characterization of the grid spacing is given which guarantees the preservation of these properties for functions in certain Lipschitz classes.