Gluing rule for Bernstein polynomials on the symmetric interval

We study special laws that arise in a sequence of the Bernstein polynomials on a symmetric interval. In particular, we set the exact rule of regular pairwise coincidence (gluing rule) which is acting for the Bernstein polynomials of a piecewise linear generating function with rational abscissas of break points. The accuracy of this rule for convex piecewise linear generating functions is shown. The possibility of “random” gluing for the Bernstein polynomials in a non-convex case is noted. We give also some examples and illustrations.