What framework/tool to use for building web based mobile apps with support for push notifications to device?
In this case you need a wrapper framework with this ability. Use any framework like jQuery mobile, Sencha touch ... and wrap it with Phonegap/Cordova framework.
Phonegap (depending on mobile platform) has a plugin to interact with push notifications.
And this is a plugin.
Google Phonegap if you need more information. It is good to know, phonegap is frameworks old name, new name is Cordova. You can use it on any mobile platform.
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