China
Africa
Explore chapters and articles related to this topic
Parametric Representations of Functions to be Estimated
Published in Zhengming Wang, Dongyun Yi, Xiaojun Duan, Jing Yao, Defeng Gu, Measurement Data Modeling and Parameter Estimation, 2016
Zhengming Wang, Dongyun Yi, Xiaojun Duan, Jing Yao, Defeng Gu
Note that functions in C[a,b] can be easily transformed into functions in C[0,1] through a transformation of the independent variable. Theorem 2.1 has a universal value of applications. The conclusion of Theorem 2.1 that a continuous function can be approximated by a polynomial at any accuracy is first proved by Weierstrass and is improved by Bernstein who provided a constructive proof using Bernstein polynomials [3]. Theorem 2.1 combines works of both Weierstrass and Bernstein.
Metric Spaces
Published in James K. Peterson, Basic Analysis III, 2020
Exercise 2.7.13Prove the polynomials with rational coefficients are dense in the set of continuous functions on [a, b] with the infinity norm. We know we can approximate any continuous function to within a given ϵ > 0 in the infinity norm using a Bernstein polynomial. The proof thus requires you to prove you can approximate any polynomial with a polynomial with rational coefficients to any accuracy desired.
Representing a Surface Using Nonuniform Rationalized B-Spline
Published in Buntara S. Gan, Condensed Isogeometric Analysis for Plate and Shell Structures, 2019
The basis function shown in Equation (1.8) is a series of curve elements (see Figure 1.8) which are used to construct a particular curve. We can see that in Figure 1.10 (n = 2), the number of basis functions is equal to the order of polynomial of the curve plus one. Equation (1.8) defines the Bernstein polynomial that is used as the basis function for constructing the Bézier curve.
A polynomial approximation-based approach for chance-constrained optimization
Published in Optimization Methods and Software, 2019
Furthermore, we also address issues such as discontinuity and non-differentiability associated with Monte Carlo. Since we adopt the Monte Carlo to approximate the cumulative distribution function of ξ, the resulting function may be discontinuous and troublesome. In addition, even if the distribution of ξ is log-concave and continuous, we still cannot assume the differentiability of . We justified all these issues mathematically. For the issue of discontinuity, we impose a minimum distance between two adjacent points to rule out the chance of discontinuity. By applying the Bernstein theorem, as long as is continuous, we can approximate it by a high-order Bernstein polynomial with arbitrary accuracy. Given the fact that the polynomial is always differentiable, in the case of non-differentibility, we can simply replace the original function with its Bernstein polynomial and assume the differentiability.
A quantized approach for occupancy grids for autonomous vehicles: Q-Trees
Published in Advanced Robotics, 2018
With the intent of simplifying things and canalizing the effort on gridding problem, quadratic and Bezier curves [19] are implemented to demonstrate a partitioned steerable path in ROI area. Bezier curves are the special implementation of Bernstein polynomials, which provides a practical way to approximate continuous functions. Bernstein polynomials are defined by where n is the number of control points that defines the curve is either quadratic (n=2) or cubic (n=3). Variable x defines the step size with and the index variable is defined with . A linear combination of Bernstein polynomials can be shown as follows: where is called Bernstein coefficient or Bezier coefficient in our terminology and substituted by which are called as Bezier control points in our case. The points are used to shape the Bezier curvature to navigate without collision. Algorithmic explanation for path association for quadratic or cubic splines is explained in Section 4.1.