Monotone approximation by algebraic polynomials

Authors:
G. G. Lorentz and K. L. Zeller

Journal:
Trans. Amer. Math. Soc. **149** (1970), 1-18

MSC:
Primary 41.40

DOI:
https://doi.org/10.1090/S0002-9947-1970-0285843-5

MathSciNet review:
0285843

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Abstract: A given real continuous function *f* on [*a, b*] is approximated by polynomials of degree *n* that are subject to certain restrictions. Let be given integers, , given signs. It is assumed that has the sign of . Theorems are obtained which describe the polynomials of best approximation, and (for ) establish their uniqueness. Relations to Birkhoff interpolation problems are of importance. Another tool are the sets *A*, where attains its maximum, and the sets with . Conditions are discussed which these sets must satisfy for a polynomial of best approximation for *f*. Numbers of the points of sets *A*, are studied, the possibility of certain extreme situations established. For example, if , it is possible that .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1970-0285843-5

Keywords:
Monotone approximation,
polynomials of best approximation,
admissible sets,
minimal polynomials,
basic polynomials of Lagrange interpolation,
Birkhoff interpolation,
free incidence matrices,
poised incidence matrices

Article copyright:
© Copyright 1970
American Mathematical Society