No, they don't. Considering 99.99% of them lie, cheat, and steal the whole damn time (and for multiple decades on their way into said role). Ya know, like appointing Supreme Court "judges" that do their bidding (eg. Bush Jr's 2nd term) and all that.
No, there've been very few politicians that actually represent their constituency, and even the rare few that do, don't tend to do so for nearly long enough.
They do not represent us, they simply shill for their handlers.