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Bernstein Polynomials by Kenneth Ijoy PDF Form

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The purpose of this short module is to prove a lemma that captures all the basic Properties of Bernstein Polynomials that will be required in the proof of the Weierstrass approximation theorem. So, let me just state the lemma. Lemma, we have number one summation K equals 0 to n B K, n of x equal to 1. We have already seen a proof of this basic property. Number two. Summation K equals 0 to n K B K, n of x is nothing, but nx and number three summation K equals 0 to n K into K minus 1 into B K, n of x is equal to n into n minus 1 into x square. So, these are some of the basic algebraic identities involving the Bernstein polynomials that will be needed in the proof of Weierstrass approximation theorem. So, let us see the proof. So, for this proof I would need one basic property of these combinations that is I need this fact that K times n choose K is just n times n minus 1 choose K minus 1. So, I am not going to prove this basic fact, it is just involves expanding out the left-hand side an

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