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- Jan 26, 2012

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Thanks to those who participated in last week's POTW!! Here's this week's problem.

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\[(f\ast g)(t) = \int_0^{t}f(t-\tau)g(\tau)\,d\tau.\]

Establish the commutative, distributive, and associative properties of convolution, i.e.

(1) $f\ast g = g\ast f$

(2) $f\ast (g_1 + g_2) = f\ast g_1 + f\ast g_2$

(3) $f\ast(g\ast h) = (f\ast g)\ast h$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!

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**Problem**: Recall that the convolution of $f$ and $g$ is defined by the integral\[(f\ast g)(t) = \int_0^{t}f(t-\tau)g(\tau)\,d\tau.\]

Establish the commutative, distributive, and associative properties of convolution, i.e.

(1) $f\ast g = g\ast f$

(2) $f\ast (g_1 + g_2) = f\ast g_1 + f\ast g_2$

(3) $f\ast(g\ast h) = (f\ast g)\ast h$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!

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