You could try, but I suspect it will to problems if you use the mobile DOCTYPE on a computer. I think you should have a sub-directory redirect. But of course, try the mobile DOCTYPE first on both platforms, and also try the XHTML Strict in the same scenario, as I am not too sure when it comes to this. Sorry.
The man who comes back through the Door in the Wall will never be quite the same as the man who went out.
Because you can't properly design a website for a desktop and mobile device in one - because of the huge difference in screen size f.e. - this is what I am always doing:
First I design the desktop website, mostly with a XHTML 1.0 Strict DOCTYPE <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">, but there is nothing wrong with using your HTML 4.01 Strict DOCTYPE either.
Then, after I have finished the desktop website, I design a mobile website and put it for example on a subdomain or dotMobi domain (m.domain.com or domain.mobi). For this mobile website I use the following DOCTYPE: <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML Basic 1.1//EN" "http://www.w3.org/TR/xhtml-basic/xhtml-basic11.dtd">
Then you can forward any mobile device to the mobile version or you can put a link to the mobile version on the desktop version. Although I haven't tested this script, it seems like this one will do: http://detectmobilebrowsers.mobi/.
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