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On the L1-Condition Number of the Univariate Bernstein Basis - Lyche, Scherer (2002) (Correct)
....are simple and located in ( 1, 1) This implies that the BB coefficients must alternate in sign. The main tools for showing these results are the variation diminishing property and the fact that collocation matrices formed from subsets of the Bernstein basis functions have positive determinants [12]. In Section 5, we use the alternating sign property of the BB coefficients to derive a dual functional representation of (1.6) In particular, we settle the question of uniqueness of extremals by relating (1.6) to a best approximation problem in the L 1 norm from a certain Chebyshev space of ....
....0 has been chosen to be minimal. Next, pick some integers j 0 , j m and set X = B , B j m . Since the elements of X are linearly independent, there is a unique polynomial span X which has zeros at the m sign changes of f and is equal to one at some other point in [ 1, 1] Since [12] X is a Chebyshev system g can have no other zeros in [ 1, 1] By the variation diminishing property of X [12, Sec. 4.9] the BB coefficients ( of must all be nonzero and alternate in sign. We normalize g so that c j,0 0. Set c j k sign(d i 0 ) c j k for k and ....
L. L. SCHUMAKER (1981): Spline Functions: Basic Theory. New York: Wiley.
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