# Transformation of Chebyshev–Bernstein Polynomial Basis

@inproceedings{Rababah2003TransformationOC, title={Transformation of Chebyshev–Bernstein Polynomial Basis}, author={Abedallah Rababah}, year={2003} }

Abstract In this paper, we derive a matrix of transformation of Chebyshev polynomials of the first kind into Bernstein polynomials and vice versa. We also study the stability of these linear maps and show that the Chebyshev–Bernstein basis conversion is remarkably well-conditioned, allowing one to combine the superior least-squares performance of Chebyshev polynomials with the geometrical insight of the Bernstein form. We also compare it to other basis transformations such as Bernstein-Hermite…

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## 46 Citations

### Generalized Chebyshev I-Bernstein bases transformation

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An explicit closed from is provided of The generalized Chebyshev-I polynomials of degree r = n in terms of the Bernstein basis of fixed degree n.

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This paper characterize the generalized Chebyshev-type polynomials of the first kind, and provides a closed form of the constructed poynomials in term of the Bernstein polynomsials.

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In this paper, the change of bases transformations between the Bernstein polynomial basis and the Chebyshev polynomial basis of the fourth kind are studied and the matrices of transformation among…

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This paper formulates a new explicit expression for the generalized Jacobi polynomials (GJPs) in terms of Bernstein basis and establishes and proves the basis transformation that embeds the perfect least-square performance of the GJPs with the geometrical insight of the Bernstein form.

## References

SHOWING 1-10 OF 11 REFERENCES

### On the optimal stability of the Bernstein basis

- MathematicsMath. Comput.
- 1996

We show that the Bernstein polynomial basis on a given interval is optimally stable, in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or…

### On the stability of polynomial transformations between Taylor, Bernstein and Hermite forms

- MathematicsNumerical Algorithms
- 2005

The stability of transformations between Taylor and Hermite and Bernstein and Hermites forms of the polynomials are investigated and an exact asymptotic is given for the condition numbers in thel1 case.

### Basis conversion among Bézier, Tchebyshev and Legendre

- Computer Science, MathematicsComput. Aided Geom. Des.
- 1998

### The dual basis functions for the Bernstein polynomials

- MathematicsAdv. Comput. Math.
- 1998

An explicit formula for the dual basis functions of the Bernstein basis is derived and they are expressed as linear combinations of Bernstein polynomials.

### Fundamentals of computer aided geometric design

- Computer Science
- 1996

A classic reference and text, this book introduces the foundations used to create an accurate computer screen image using mathematical tools. This comprehensive guide is a handbook for students and…