7.5 Total Variation and the Variation Diminishing Property

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 $V(f,[a,b])$ be the total variation of $f$ over $[a,b]$ and let $f: [0,1]\to\mathbb{R}$ be a continuous function. Then

$$V(B_{f,n},[0,1]) \le V(f,[0,1]),$$

where the equality sign holds if and only if the function $f$ 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 $Z(f,(a,b))$ be the number of real zeros of $f$ in the interval $(a,b)$ and let $f: [0,1]\to\mathbb{R}$ be a continuous function. Then

$$Z(B_{f,n},(0,1)) \le v(f),$$

where $v(f)$ is the number of changes of sign of $f$ in $[0,1]$.

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.