Shape preserving representations for trigonometric polynomial curves

This paper has two main goals. Firstly, we show that the space of trigonometric polynomials tau(m) = span(1, cos t, sin t,..., cos mt, sin mt) is not suitable for those methods of CAGD which use control polygons. It is well-known that the bases with good shape preserving properties are the normalized totally positive bases and we prove here that tau(m), does not possess normalized totally positive bases. Secondly, we show that the space C-m = span(1, cos t,..., cos mt) is suitable for design purposes using control polygons. In fact, we construct a basis C-m of C-m with optimal shape preserving properties and analyze some aspects for the computation of the corresponding curves.