Solving change of basis from Bernstein to Chebyshev polynomials

We provide two closed-form solutions to the change of basis from Bernstein polynomials to shifted Chebyshev polynomials of the fourth kind and show them to be equivalent by applying Zeilberger’s algorithm. The first solution uses orthogonality properties of the Chebyshev polynomials. The second is “...

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Vydané v:Examples and counterexamples Ročník 7; s. 100178
Hlavný autor: Wolfram, D.A.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Elsevier B.V 01.06.2025
Elsevier
Predmet:
Change of basis
Connection formula
Polynomial basis
secondary
Connection formula
Change of basis
primary
Polynomial basis
ISSN:2666-657X, 2666-657X
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Shrnutí:We provide two closed-form solutions to the change of basis from Bernstein polynomials to shifted Chebyshev polynomials of the fourth kind and show them to be equivalent by applying Zeilberger’s algorithm. The first solution uses orthogonality properties of the Chebyshev polynomials. The second is “modular” which enables separately verified sub-problems to be composed and re-used in other basis transformations. These results have applications in change of basis of orthogonal, and non-orthogonal polynomials.
ISSN:2666-657X
2666-657X
DOI:10.1016/j.exco.2025.100178