1he convolution of two functions is defined by fgx

Question

1. The convolution of two functions is defined by, (f*g)(x) = 1/√2? ∫ f(s) g(x-s) ds (integral is from -∞ to ∞)

a) Show by differentiation under the integral that if g is differentiable then (f*g)(x) is differentiable with (f*g)'(x) = (f*g')(x)

b) Show that for t>0 the fundamental solution to the heat equation, φ(x,t) = (1/c√2t) e^(-x²/4c²t) has a continuous special derivative. (i.e. Differentiate with respect to x then argue why its continuous for t>0)

c) Using parts a and b, argue that the solution to the Heat Equation, ∂u/∂t = c²(∂²u/∂x²) -∞ < x < ∞ u(x,0) = f(x) is differentiable for t>0 even if f is not differentiable.