Interpolation and Extrapolation

Given a set of discrete data points representing the known values $v({\bf x}_i)$ at certain positions ${\bf x}_i,\;(i=1,\cdots,n)$ in an n-D space, we may want to estimate the value $v({\bf x})$ at any other point ${\bf x}$, using the method of interpolation if ${\bf x}$ is inside the range of the known data points, or extrapolation if ${\bf x}$ is outside the range. For exammple, the known values of data points may be a set of measuremnts (samples) of certain variable (e.g.,temperature) over space (3-D) and time (1-D), and the goal is to estimate its value at any other point in space and time. Typically this can be achieved by fitting the dataset by a continuous function $f({\bf x})$, such as a polynomial, that either passes through the data points $f({\bf x}_i)=v({\bf x}_i)$, or approximates the data ponts $f({\bf x}_i)\approx v({\bf x}_i)$. The function $f({\bf x})$ should be continuous ($C^0$ continuous) and preferably smooth ($C^k$ continuous with $k>0$).