Shape Preserving Approximation of Periodic Functions: Conclusion

We give here the final results about the validity of Jackson-type estimates in comonotone approximation of \(2\pi \) $2\pi$ -periodic functions by trigonometric polynomials ( \(q=1\) $q=1$ ). For coconvex ( \(q=2\) $q=2$ ) and the so called co-q-monotone, \(q>2\) $q>2$ , approximations, everything is known by now. Thus, this paper concludes the research on Jackson type estimates of Shape Preserving Approximation of periodic functions by trigonometric polynomials. It is interesting to point out that the results for comonotone approximation of a periodic function are substantially different than the analogous results for comonotone approximation of a continuous function on a finite interval, by algebraic polynomials.