Problem 4.4 Given two functions f and g their convolution f ∗ g is the function defined by the formula:

[f ∗ g](x) =

 +∞

−∞

f(y)g(x − y) dy 1. Use a simple substitution to check that f ∗ g = g ∗ f.

2. Prove that:

Ktriangle = Kbox ∗ Kbox and that:

Kparzen = Ktriangle ∗ Kbox.

Recall the definitions of these kernels given in Table 4.5.

3. Use S-Plus to plot the four kernel functions given in Table 4.5. Explain your work, and try to create the plots without using any mathematical formula.