Assume that the real-valued function f(x) on the interval [a,b] has n+1 different points x0,x1,……,xn in the interval. The value at xn is f (x0),……f(xn ), it is required to estimate the value of f(x) at a certain point x* in [a,b]. The basic idea is to find a function P(x) that has the same value as the function f(x) at the nodes of x0, x1,..., xn (sometimes, even the first derivative value is the same), use P(x*) The value of is used as an approximation of the function f(x*).
The usual approach is: in a pre-selected simple function composed of n+1 parameters C0, C1, ... Cn function class Φ (C0, C1, ... Cn) to find the condition P( xi)=f(xi)(i=0,1,……n) function P(x), and use P() as the evaluation of f(). Here f(x) is called the interpolated function, x0, x1,..., xn is called the interpolation node (node) point, Φ(C0, C1,...Cn) is called the interpolation function class, and the above equation is called Interpolation conditions, the function that satisfies the above formula in Φ(C0, C1,...Cn) is called an interpolation function, and R(x) = f(x)-P(x) is called an interpolation remainder. When the estimated point belongs to the smallest closed interval containing x0, x1,..., xn, the corresponding interpolation is called interpolation, otherwise it is called extrapolation.