Polynomial Approximations

Consider a function of time f( t) whose values are given at the ( n + 1) distinct points t ₀, t ₁,..., t _n. It is always possible to find an interpolating polynomial of degree n which passes exactly through these ( n + 1) points. Let us choose

(6.1)

as the approximating polynomial to y = f( t) and let

(6.2)

We can determine the ( n + 1) coefficients a ₀, a ₁,..., a _n from the ( n + 1) equations

(6.3)

which are linear in the as. For distinct times t ₀, t ₁,..., t _n, a solution for the as is always possible because the determinant of their coefficients is always nonzero. Thus, we can obtain the interpolating polynomial P _n( t).

Lagrange's Interpolation Formula

An alternative form of the same interpolating polynomial can be obtained by first using the notation

(6.4)

We note that the factor ( t - t _i) is omitted. Now define a sampling polynomial

(6.5)

which is a polynomial of degree n in t. It has the property that it is equal to zero at all the sampling times t ₀, t ₁, . . . except at t _i where its value is one. It is apparent, then, that a polynomial of degree n which...