The following theorem states that the total variation of the Bernstein
polynomial of a function of one variable is less equal than the total
variation of the function itself. Thus the Bernstein approximation
operator has a smoothing effect.

**Theorem 7..14** (Total Variation)
Let

be the total variation of

over

and let

be a continuous function. Then

where the equality sign holds if and only if the function

is
monotone.

This means the approximation is smoother than the original function
regarding the amount of total variation. Proofs of this theorem can
be found in [96] and [109], where the case of
equality is discussed.
Not only is the total variation reduced by the Bernstein operator, but
they also have the following variation diminishing property.

**Theorem 7..15** (Variation Diminishing Property)
Let

be the number of real zeros of

in the interval

and let

be a continuous function. Then

where

is the number of changes of sign of

in

.

This last theorem is the reason for the excellent smoothing properties
of polynomials of Bernstein type. It states that Bernstein
polynomials should be used whenever a polynomial approximation is
needed which does not oscillate more often about any straight line
than the function to be approximated [109].
Concerning the numerical aspect, an implementation for univariate
Bernstein polynomials was presented in [144]. The higher the
degree of the approximation polynomial, the more care has to be taken
in their numerical evaluation. In the cases needed for our
applications, this is not an issue.

Clemens Heitzinger
2003-05-08