approximation by polynomials

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.

An important issue here is keeping the number of points interpolated as low as possible. As the degree of the interpolating polynomial increases, it loses smoothness, and often results in an awful approximation in all but the interpolated points.

A common solution is to approximate different intervals of the interpolated function with different polynomials of low degree, and to impose continuity and differentiability conditions on the whole. Different formulas give us different kinds of curves, like splines, bezier curves, etc.