I would do this in two steps:
First step complete an sql query getting a summary with a sum for weight with a group by for yearweek
SELECT Company_Name, YEARWEEK(Date), sum(weight) FROM table GROUP BY Company_Name, YEARWEEK(Date)
http://dev.mysql.com/doc/refman/5.5/en/date-and-time-functions.html#function_yearweek.
Second step would be to process this into the required format in the application year.
If you absolutely have to do this in the database, then you are looking at implementing a pivot table, which has previously been covered here: MySQL pivot table