The Fourier Transform on R 4 Bump functions Examples of comp

The Fourier Transform on R:

4. Bump functions. Examples of compactly supported functions in S(R) are very handy in many applications in analysis. Some examples are (a) Suppose a b, and f is the function such that f(z) 0 if z s a or z 2 b an 1/(b-r) f(z) if a b Show that f is indefinitely differentiable on R. (b) Prove that there exists an indefinitely differentiable function F on R such that F(z) 0 if a S a, F(ar) 1 if b, and F is strictly increasing on a, b] (c) Let 0 be so small that a

Solution

f(x)=e^-1/(x-a) . e^-1/(b-x)

Taking differentiation we get

f\'(x) = e^-1/(x-a). e^-1/(b-x) (-1/(b-x)) + e^-1/(b-x) .e^-1/(x-a)(-1/x-a).

= e^-1/(x-a). e^-1/(b-x) (-1/b-x + -1/x-a)

= e^-1/(x-a). e^-1/(b-x) ( x-a-x-b/(x-a)(x-b))

= e^-1/(x-a). e^-1/(b-x) ( -a-b/(x^2+ab-2ab))

for different values of x function behaves differently

so from obtained eqution f is indifinitely differentiable on R

From the obtained solution as there are 2 variables a and b.

taking different values we get different solutions.