## Uniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials

- Jan. 7, 2011
- 2:30 p.m.
- LeConte 312

## Abstract

We survey the known Jackson type estimates in the Shape Preserving Approximation (SPA), of functions in C[-1, 1], by algebraic polynomials. Comparing these estimates with what is known for unconstrained approximation and pointing out many open problems. In those cases in which we have information on the validity and invalidity of the Jackson estimates, the answers to that range from strongly positive, through weakly positive, weakly negative to strongly negative. We also investigate pointwise estimates and again discuss their validity and invalidity. Here too, we have the above mentioned four possibilities, but we encounter a fifth possibility. Again, there are many open questions.