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.

(load "43-repeated.scm")

(define (smooth f)
  (define dx 0.00001)
  (lambda (x)
    (/ (+ (f (+ x dx))
          (f x)
          (f (- x dx)))
       3)))

(define (n-fold-smooth-f n f)
  (lambda (x)
    ((repeated smooth n) f) x))

;;; todo: add tests