  Maple worksheets on Bernstein polynomials and polynomial approximation  Approximation of functions:

They are all compatible with Classic Worksheet Maple 10. BEZIER CURVE Bezier curves and Bernstein polynomials  - bezier.mws

• Introduction
• 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

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