Bernstein polynomial

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The [ilmath]n[/ilmath]th Bernstein polynomial for a function [ilmath]f:I\rightarrow\mathbb{R} [/ilmath] (where [ilmath]I:\eq[0,1]\subset\mathbb{R} [/ilmath] is the closed unit interval) is defined as follows:

  • [math]\mathcal{B}_n(f;x):\eq\sum^n_{i\eq 0}{}^nC_if\left(\tfrac{k}{n}\right)x^i(1-x)^{n-i} [/math]


Notice that the binomial expansion of [ilmath](x+(1-x))^n\eq 1^n\eq 1[/ilmath] is [ilmath]\sum^n_{i\eq 0}{}^nC_ix^i(1-x)^{n-i} [/ilmath] so we see: [ilmath]\sum^n_{i\eq 0}{}^nC_ix^i(1-x)^{n-i}\eq 1[/ilmath] provided [ilmath]\vert x\vert < 1[/ilmath] and [ilmath]\vert 1-x\vert < 1[/ilmath], we also note that if [ilmath]x\eq 0[/ilmath] or [ilmath]x\eq 1[/ilmath] then it is also equal to 1

TODO: More analysis required