OK, here's my attempt. It steals ideas from the other answers, and attempts to make the logic easier to follow. This is based on the ISO 8601 system, so it won't be correct if you live in countries such as USA or Canada, but should be easily adjustable for those countries.
# sets $week_start to a representation of Monday of the given week
# number formatted via the given format, and similarly sets
# $week_end to Friday of the same week.
get_week_range () {
week_num="$1" date_format="$2"
# Most of the world adhere to ISO 8601 which states that weeks begin on Monday
# and Jan 4th is always in week #1:
#
# http://en.wikipedia.org/wiki/ISO_week_date
#
# For other week numbering systems (e.g. USA, Canada), see:
#
# http://en.wikipedia.org/wiki/Seven-day_week#Week_numbering
day_in_week_1=$( date +'%Y-01-04' )
day_num_in_week_1=$( date -d $day_in_week_1 +%u ) # 1 is Monday
days_from_week_1_start=$(( $day_num_in_week_1 - 1 ))
# This is a Monday:
start_of_week_1=$( date -d "$day_in_week_1 - $days_from_week_1_start days" +%F )
week_delta="$(( $week_num - 1 ))"
# Monday:
week_start=$( date -d "$start_of_week_1 + $week_delta weeks" +"$date_format" )
# Friday:
week_end=$( date -d "$start_of_week_1 + $week_delta weeks + 4 days" +"$date_format" )
}
%V
format sequence used by 'user unknown' reports the ISO week number. – Peter.O Sep 01 '11 at 16:16