Exercise 1.44. The idea of smoothing a function is an important concept in signal processing.
If f is a function and dx is some small number, then the smoothed version of f is the function whose value at a point x is the average of f(x – dx), f(x), and f(x + dx).
Write a procedure smooth that takes as input a procedure that computes f and returns a procedure that computes the smoothed f. It is sometimes valuable to repeatedly smooth a function (that is, smooth the smoothed function, and so on) to obtained the n-fold smoothed function. Show how to generate the n-fold smoothed function of any given function using smooth and repeated from exercise 1.43.