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Degenerate Bernstein polynomials. (English) Zbl 0534.41005

For \(f\epsilon\) C[0,1], the n-th Bernstein polynomial \(B_ n(f;x)\) is a polynomial of exact degree n, although degeneracies can occur in some cases. For example, if f itself is a polynomial of degree m, then \(B_ n(f;x)\) is also of degree m for \(n\geq m\) (although not equal to f(x) except in the case \(m=1)\). The result can be verified from an alternate form of \(B_ n(f;x).\)
In the present note, the authors study a class of functions which has a sequence of degenerate Bernstein polynomials. Moreover, a surprising result on an equality for certain of these polynomials is established.
Reviewer: V.Singh

MSC:

41A10 Approximation by polynomials
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References:

[1] Lorentz, G. G., Bernstein Polynomials (1953), Univ. of Toronto Press: Univ. of Toronto Press Toronto · Zbl 0051.05001
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