Using Google Analytics, how can we see how many new visitors from last week came back this week?
Disclaimer: This is just top of the hat and I have not really tested this, yet.
The Approximate Approach
If you are OK with total returning visitors this week as compared to the new visitors last week, then it is fairly simple.
This might not be the exact solution which you are seeking for (since the returning visitors in the current period are not necessarily attributed only to the new visitors of last week, thats a subset of ALL the new visitors who have returned).
The Programming Approach
An alternative method is to add a Custom Variable for all the new visitors coming to the site, for this you will have to do a bit of cookie parsing. You need to parse the __utma cookie of GA and here's where you will find the structure of the cookies (you need to take the inital visit timestamp, compare it to today's date and if it matches, then this is a new visitor, set a custom variable for this visitor, let's say Week and assign it the value of the week number in the year (1 to 54).
Now that you have tagged New Visitors with the week number, all you have to do is -
Again, this can only work once you have done the programming bit and not for the data that already exists in your GA.