Let x_(0)lt x_(1)lt ... lt x_(N) be N+1 distinct interpolation points. Prove that sum _(i=0)^NL_(i)(x)=1 where L_(i)(x) for i=0,1,... ,N are the Lagrange basis polynomials.

Question

Answer 1

To prove that

, we can use the properties of the Lagrange basis polynomials.The Lagrange basis polynomials are defined as follows:

Now, let's consider the sum

:

We can rewrite this sum as:

Notice that the sum

is equal to 1, since it is a telescoping sum.Therefore, we have:

Hence, we have proved that

.

Soru