Where can we find examples of smooth functions with compact support? Is there a book?

Question

Where can we find examples of smooth functions of compact support? Is there a book? How to show that $$ f(x)=\exp\left(-\frac{1}{x} \right) \exp\left(-\frac{1}{1-x}\right) $$ is smooth?

Answer 1

It should be $$ f(x) = \exp \left( \frac{-1}{x} \right) \exp \left( \frac{1}{x-1} \right) , \; \; \; 0 < x < 1 $$ and $f(x) = 0$ for all other $x$. Smoothness follows from smoothness for $\exp (-1/x), x > 0.$

Next, there is some constant $A = \int_0^1 f(x) dx.$ We get an almost step function by $$ g(x) = \frac{1}{A} \int_{- \infty}^x f(t) dt $$

Then we get a plateau from $$ h(x) = g(x) g(B - x) $$ for some $B > 1.$

Answer 2

Here's an explanation for why the function you gave is smooth. Consider the function $f(x) = \exp(-1/x)$.

Step 1: Show by induction that $f^{(n)}(x) = p_n(1/x) \exp(-1/x)$ for some polynomial $p_n(x)$.

Step 2: Show that $\lim_{x \to 0^+} p(1/x) e^{-1/x} = 0$ for any fixed polynomial $p(x)$.

Step 3: If we define a function $g(x) = 0$ for $x \leq 0$ and $g(x) = f(x)$ for $x>0$, then on $(0,\infty)$ and $(-\infty,0)$ the $n$th derivatives clearly exist and are continuous. You must show that the left and right derivatives of $g^{(n)}(x)$ definitionally are both zero at $x=0$, then, to conclude that $f(x)$ is smooth. Use step 2.

Step 4: Finally, the function $g(x)g(1-x)$ is equivalent to the function you asked about (I think you made a typo) and is smooth since it is made up of smooth functions. It has compact support by inspection.

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For other examples of smooth, compactly support functions, consider the convolution of any compactly supported function with a smooth compactly supported function. That is, let $\phi(x) \in C_c^\infty(R)$ and let $f(x)$ and $\phi(x)$ have compact support in $B_r(x)$. Then

$$g(x) = (f * \phi)(x) = \int_{-\infty}^{\infty} f(x-y)\phi(y) dy$$

is both smooth and compactly supported. (See the wikipedia article on mollifiers for similar examples.)