Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

Abstract

We study the existence and shape preserving properties of a generalized Bernstein operator $B_{n}$ fixing a strictly positive function $f_{0}$, and a second function $f_{1}$ such that $f_{1}/f_{0}$ is strictly increasing, within the framework of extended Chebyshev spaces $U_{n}$. The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator $B_{n}:C[a,b]\to U_{n}$ with strictly increasing nodes, fixing $f_{0}, f_{1}\in U_{n}$. If $U_{n}\subset U_{n + 1}$ and $U_{n + 1}$ has a non-negative Bernstein basis, then there exists a Bernstein operator $B_{n+1}:C[a,b]\to U_{n+1}$ with strictly increasing nodes, fixing $f_{0}$ and $f_{1}.$ In particular, if $% f_{0},f_{1},...,f_{n}$ is a basis of $U_{n}$ such that the linear span of $% f_{0},..,f_{k}$ is an extended Chebyshev space over $[ a,b] $ for each $k=0,...,n$, then there exists a Bernstein operator $B_{n}$ with increasing nodes fixing $f_{0}$ and $f_{1}.$ The second main result says that under the above assumptions the following inequalities hold B_{n}f\geq B_{n+1}f\geq f for all $(f_{0},f_{1})$-convex functions $f\in C[ a,b] .$ Furthermore, $B_{n}f$ is $(f_{0},f_{1})$-convex for all $(f_{0},f_{1})$% -convex functions $f\in C[ a,b] .$ In the specific case of exponential polynomials we give alternative proofs of shape preserving properties by computing derivatives of the generalized Bernstein polynomials.