Editing Bernstein polynomial (section)

== Generalizations to higher dimension ==
Bernstein polynomials can be generalized to {{math|''k''}} dimensions – the resulting polynomials have the form {{math| ''B''<sub>''i''<sub>1</sub></sub>(''x''<sub>1</sub>) ''B''<sub>''i''<sub>2</sub></sub>(''x''<sub>2</sub>) ... ''B''<sub>''i''<sub>''k''</sub></sub>(''x''<sub>''k''</sub>)}}.<ref name="Lorentz"/> In the simplest case only products of the unit interval {{math|[0,1]}} are considered; but, using [[affine transformation]]s of the line, Bernstein polynomials can also be defined for products {{math|[''a''<sub>1</sub>, ''b''<sub>1</sub>] × [''a''<sub>2</sub>, ''b''<sub>2</sub>] × ... × [''a''<sub>''k''</sub>, ''b''<sub>''k''</sub>]}}. For a continuous function {{math|''f''}} on the {{math|''k''}}-fold product of the unit interval, the proof that {{math|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ... , ''x''<sub>''k''</sub>)}} can be uniformly approximated by

:<math>\sum_{i_1} \sum_{i_2} \cdots \sum_{i_k} {n_1\choose i_1} {n_2\choose i_2} \cdots {n_k\choose i_k}
f\left({i_1\over n_1}, {i_2\over n_2}, \dots, {i_k\over n_k}\right) 
x_1^{i_1} (1-x_1)^{n_1-i_1} 
x_2^{i_2} (1-x_2)^{n_2-i_2} 
\cdots x_k^{i_k} (1-x_k)^{n_k - i_k}
</math>
  
is a straightforward extension of Bernstein's proof in one dimension.
<ref>{{citation|last1=Hildebrandt|first1= T. H.|authorlink=Theophil Henry Hildebrandt|last2=Schoenberg|first2= I. J.|authorlink2= I. J. Schoenberg|title= On linear functional operations and the moment problem for a finite interval in one or several dimensions|journal=[[Annals of Mathematics]]|volume= 34|year=1933|issue= 2|page=327|doi= 10.2307/1968205|jstor= 1968205|url=https://www.jstor.org/stable/1968205}}</ref>