Maple worksheets on Bernstein polynomials and polynomial approximation |

Approximation of functions:

- Taylor Series
- Fourier Series
- Convergence of Fourier Series
- Bernstein polynomials and polynomial approximation

The following Maple worksheets can be downloaded.

They are all compatible with

Classic Worksheet Maple 10.

BEZIER CURVE

Bezier curves and Bernstein polynomials- bezier.mws

- Introduction
- Quadratic Bezier curves
- Cubic Bezier curves
- General Bezier curves and Bernstein polynomials
- An animation procedure for Bezier curves and Bernstein polynomials:
bezier.

The Bernstein basis polynomials and de Casteljau's algorithm- casteljau.mws

- Bernstein basis polynomials
- A recursive definition of the Bernstein polynomials
- An inductive proof that the Bernstein polynomials form a partition of unity
- de Casteljau's algorithm
- A procedure implementing de Casteljau's algorithm:
casteljau.

Bernstein polynomials associated with powers of x- brnpower.mws

- Bernstein polynomials for 1
- Bernstein polynomials for x
- Bernstein polynomials for x-x^2
- Bernstein polynomials for x^2
- Bernstein polynomials for x-x^3
- Bernstein polynomials for x^3
- Summary

The Bernstein polynomials as a basis for polynomials- brnbasis.mws

- The degree raising formula
- Expressing a power x^k as a linear combination of Bernstein polynomials
- The Bernstein polynomials as a basis
- Matrices for the change of basis
- The Bernstein coefficients of the Bernstein polynomial associated with a polynomial function

The Weierstrass polynomial approximation theorem- weierstrass.mws

- The Weierstrass theorem
- Reduction to continuous functions on [0,1]
- The modulus of continuity of a continuous functions on [0,1]
- A constructive proof of the Weierstrass theorem based on Bernstein polynomials

Approximation of functions using partitions of unity- partunit.mws

- Introduction - Bernstein polynomials
- Approximation using rectangular pulses
- Approximation using "bell-shaped" pulses"
- Approximation using wider "bell-shaped" pulses"

More examples of partitions of unity- partunit2.mws

- Approximation using "bell-shaped" pulses" - II
- Approximation using wider "bell-shaped" pulses" - II

Examples of Bernstein polynomials on [-1,1]- brnexamp.mws

Comparison of polynomial approximations- polyapprox.mws

- Introduction
- Taylor polynomials
- Legendre polynomials
- Bernstein polynomials
- Summary

**Function approximation procedures**
- fcnapprx.zip

BEZIER BASIS POLYNOMIALS