ABSTRACT
As we saw in the last chapter, the mere fact that we can interpolate some samples taken from a function does not necessarily mean that we have approximated the function well. In fact, the constraint of interpolation can make our “ approximation” arbitrarily bad almost everywhere except at the points we are interpolating. Approximating a function-perhaps even interpolating no points on the curve-is sometimes preferable to interpolation. There is a vast array of approximation techniques, and we cannot hope to cover many of them in depth. In this chapter we shall introduce approximation as motivated by the discrete sampling and reconstruction of functions. The topics to be discussed include:
• how many samples are needed to approximate an unknown function? • uniform vs. nonuniform sampling. • coping with noisy input data. • convolution and filtering. • applications. • computational issues.
