Onesided, intertwining, positive and copositive polynomial approximation with interpolatory constraints. (English) Zbl 1534.41009
Polynomial interpolation either in Lagrange form or in Hermite form is a standard method for univariate approximation on intervals. Especially if smooth functions of a variety of smoothness classes are approximated, many convergence and approximation order results and estimates are available.
At the same time, results are available for monotonicity preserving requirements for example and for the existence of polynomials that satisfy them. In this paper these constraints are combined in order to achieve very powerful and general results that answer questions about the existence (or not) of polynomial approximants on intervals that interpolate (satisfy Hermite interpolation conditions) and at the same time satisfy geometrical conditions. Such as non-negativity, monotonicity (the approximant is bounded by the function either ways) intertwining properties (the polynomial bounds the function alternatingly from above and below, changing at the origin if the approximation domain is \([-1,1]\)).
Among other things, the authors establish precise results about the existence of such approximations that should simultaneously satisfy error bounds of the classical type (using moduli of continuity). They also list cases where these types of polynomial interpolants do not exist. The existence of the described approximants is highly dependent on the smoothness and desired order of convergence as well as on the order of the said modulus of continuity and, of course, on the order of the polynomials that are used.
At the same time, results are available for monotonicity preserving requirements for example and for the existence of polynomials that satisfy them. In this paper these constraints are combined in order to achieve very powerful and general results that answer questions about the existence (or not) of polynomial approximants on intervals that interpolate (satisfy Hermite interpolation conditions) and at the same time satisfy geometrical conditions. Such as non-negativity, monotonicity (the approximant is bounded by the function either ways) intertwining properties (the polynomial bounds the function alternatingly from above and below, changing at the origin if the approximation domain is \([-1,1]\)).
Among other things, the authors establish precise results about the existence of such approximations that should simultaneously satisfy error bounds of the classical type (using moduli of continuity). They also list cases where these types of polynomial interpolants do not exist. The existence of the described approximants is highly dependent on the smoothness and desired order of convergence as well as on the order of the said modulus of continuity and, of course, on the order of the polynomials that are used.
Reviewer: Martin D. Buhmann (Gießen)
MSC:
| 41A25 | Rate of convergence, degree of approximation |
| 41A10 | Approximation by polynomials |
| 41A29 | Approximation with constraints |
| 41A05 | Interpolation in approximation theory |
| 41A15 | Spline approximation |
Keywords:
intertwining; copositive; positive; nonnegative; Hermite interpolation; approximation by algebraic polynomials; exact pointwise estimates; interpolatory estimates; moduli of smoothness; exact estimatesReferences:
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