We discuss whether or not it is possible to have interpolatory pointwise estimates in the approximation of a function f ∈ C[0, 1], by polynomials. For the sake of completeness, as well as in order to strengthen some existing results, we discuss briefly the situation in unconstrained approximation. Then we deal with positive and monotone constraints where we show exactly when such interpolatory estimates are achievable by proving affirmative results and by providing the necessary counterexamples in all other cases.
- Approximation by polynomials
- Degree of approximation
- Interpolatory pointwise estimates
- Shape-preserving approximation