  We introduce an algorithm of joint approximation of a function and its first
derivative by alternative orthogonal polynomials on the interval [0,1].The
algorithm exhibits properties of shape preserving approximation for the
function. A weak formulation of approximation is presented. An example on shape
preserving extrapolation is given. The weak form is reduced for approximation
on a discrete set of abscissas.
  Also, we introduce a new system of orthogonal functions with nice properties
- structured orthogonal polynomials - and show that the system can be employed
for a different kind of joint approximation of a function and its first
derivative and may have property of shape preserving approximation. In
addition, we show that structured orthogonal polynomials generate wavelet
functions
  We complement these results with definition of structured semi-orthogonal
polynomials and introduce wavelet basis functions.
