Late to the thread, but I wanted to quickly mention an identity that shows up for separable functions. Although this is a close cousin of your trivial identity and hardly theoretically deep, it turns out to be very useful in practice.

Let's take $\mathrm{R}^2$ as an example. If $f(x,y) = f_1(x)\ f_2(y)$ and $g(x,y) = g_1(x)\ g_2(y)$ then

$$f * g = (f_1\ f_2) * (g_1\ g_2) = (f_1 * g_1)\ (f_2 * g_2).$$

I abused notation a little to highlight the resemblance to distributivity. This identity finds use in a folklore trick of image processing that is described here:

http://www.stereopsis.com/shadowrect/

Edit: This is not true in general. take $$f_1=f_2=g1=g2=\Pi,$$ where $\Pi$ denotes the rectangular function, which is one in the interval $[-1/2,1/2]$ and zero elsewhere. Then: $$(f_1\ f_2)*(g_1\ g_2)=(\Pi^2)*(\Pi^2)=\Pi*\Pi=\Delta\neq \Delta^2=(\Pi*\Pi)^2=(f_1* g_1)(f_2*g_2).$$ Here, $\Delta$ is the triangle function.

Late to the thread, but I wanted to quickly mention an identity that shows up for separable functions. Although this is a close cousin of your trivial identity and hardly theoretically deep, it turns out to be very useful in practice.

Let's take $\mathrm{R}^2$ as an example. If $f(x,y) = f_1(x)\ f_2(y)$ and $g(x,y) = g_1(x)\ g_2(y)$ then

$$f * g = (f_1\ f_2) * (g_1\ g_2) = (f_1 * g_1)\ (f_2 * g_2).$$

I abused notation a little to highlight the resemblance to distributivity. This identity finds use in a folklore trick of image processing that is described here:

http://www.stereopsis.com/shadowrect/

Edit by anonymous user: This is not true in general. Take $$f_1=f_2=g_1=g_2=\Pi,$$ where $\Pi$ denotes the rectangular function, which is one in the interval $[-1/2,1/2]$ and zero elsewhere. Then: $$(f_1\ f_2)*(g_1\ g_2)=(\Pi^2)*(\Pi^2)=\Pi*\Pi=\Delta\neq \Delta^2=(\Pi*\Pi)^2=(f_1* g_1)(f_2*g_2).$$ Here, $\Delta$ is the triangle function.

Late to the thread, but I wanted to quickly mention an identity that shows up for separable functions. Although this is a close cousin of your trivial identity and hardly theoretically deep, it turns out to be very useful in practice.

Let's take $\mathrm{R}^2$ as an example. If $f(x,y) = f_1(x)\ f_2(y)$ and $g(x,y) = g_1(x)\ g_2(y)$ then

$$f * g = (f_1\ f_2) * (g_1\ g_2) = (f_1 * g_1)\ (f_2 * g_2).$$

I abused notation a little to highlight the resemblance to distributivity.

  This identity finds use in a folklore trick of image processing that is described here:

http://www.stereopsis.com/shadowrect/

Late to the thread, but I wanted to quickly mention an identity that shows up for separable functions. Although this is a close cousin of your trivial identity and hardly theoretically deep, it turns out to be very useful in practice.

Let's take $\mathrm{R}^2$ as an example. If $f(x,y) = f_1(x)\ f_2(y)$ and $g(x,y) = g_1(x)\ g_2(y)$ then

$$f * g = (f_1\ f_2) * (g_1\ g_2) = (f_1 * g_1)\ (f_2 * g_2).$$

I abused notation a little to highlight the resemblance to distributivity. This identity finds use in a folklore trick of image processing that is described here:

http://www.stereopsis.com/shadowrect/

Edit: This is not true in general. take $$f_1=f_2=g1=g2=\Pi,$$ where $\Pi$ denotes the rectangular function, which is one in the interval $[-1/2,1/2]$ and zero elsewhere. Then: $$(f_1\ f_2)*(g_1\ g_2)=(\Pi^2)*(\Pi^2)=\Pi*\Pi=\Delta\neq \Delta^2=(\Pi*\Pi)^2=(f_1* g_1)(f_2*g_2).$$ Here, $\Delta$ is the triangle function.