The smoothing property of Bernstein Polynomial, proved by Kelisky and Rivilin in 1967, $$ \lim_{k\rightarrow\infty}B\;^{(k)}\left(f;x\right)=\left(1-x\right)f\left(0\right)+xf\left(1\right) $$ can be seemingly borne out by the following result: $$ {\mathrm B}_{\mathrm n}\left(\mathrm x(1-\mathrm x);\mathrm x\right)=\left(1-\frac1{\mathrm n}\right)\mathrm x(1-\mathrm x) $$

Any concrete ideas?


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