APM346-2012 > Term Test 2

TT2--Problem 1

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Victor Ivrii:
Let $f:{\mathbb{R}}\rightarrow {\mathbb{R}}$ be a continuous but non-differentiable function which satisfies $f(x)=0$ for all $|x| > 1.$ Let $g:{\mathbb{R}}\rightarrow {\mathbb{R}}$ be a continuous function which satisfies satisfies $g(x)=0$ for all $|x| > 2.$ Suppose further that derivative $g'$ and second derivative $g''$ are both continuous. The convolution $f*g$ of these two functions is defined by the formula
$$
(f*g) (x) = \int f(x-y)g(y)\,dy.
$$

* (a) Prove that the function $f*g(x) =0$ for $|x|>3$.
* (b) Prove that the derivative of the function $f*g$ is continuous.
post after 22:30

Jinchao Lin:
Solution for part(a)

Ian Kivlichan:
Hopeful solutions to both parts attached! :)

Chen Ge Qu:
See attached

Ian Kivlichan:
Chen Ge, in 1.b) I'm not sure you can have the derivative of f since it isn't differentiable.. :S

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