approximation by polynomials (idea)

Let f:[a,b] -> R a continuous function, [a,b] an interval in R and P_n the set of polynomials of degree lesser or equal n.
Let {x_i| i in {1,...,m} } a set of points in [a,b].
A polynomial p of P_n is said to interpolate f iff f(x_i)=p(x_i) for all i of {1,...,m}. This is the most common approximation by polynomials. Other approximations are rarely used.

If m <= n+1 then a p, which interpolates f, exists and if m >= n+1 then p is unique (but doesn't have to exist !), so you would always choose m = n+1.
Note that for increasing n and m=n+1 the solutions don't have to converge to f !
They converge only for special x_i and surprisingly the point mustn't have all the same distance. They must be clustered around the ends of the interval, x_i must be equal to a + (b-a) arccos(i/m)/pi.

A simple way to calculate the interpolating p is (for m=n+1) using the Lagrange's formula:

       __m  (x-x1) ... (x-x<i-1>)(x-x<i+1>) ... (x-xm)                  
p(x) = ||   ------------------------------------------ f(xi)
       i=1  (xi-x0)...(xi-x<i-1>)(xi-x<i+1>)...(xi-xm)

But there is a number of more clever formulas to calculate p.