How to convert number of week into date?

Given you have year and cw (calender week) as variables (e.g. from a SELECT statement) you can get the DATE as following:

DATE_SUB(
  DATE_ADD(MAKEDATE(year, 1), INTERVAL cw WEEK),
  INTERVAL WEEKDAY(
    DATE_ADD(MAKEDATE(year, 1), INTERVAL cw WEEK)
  ) -1 DAY),

The phrase DATE_ADD(MAKEDATE(year, 1), INTERVAL cw WEEK) is duplicated; did not want to store a variable. The SQL-Statement worked nicely for me on MySQL.

UPDATE: Just for clarification: WEEKDAY(DATE_ADD(MAKEDATE(year, 1), INTERVAL cw WEEK)) will yield the first day of the week. Substracting a number from it (-1 for Tuesday; -2 for Wednesday and so forth will select a specific day in the week for you). See here.

In MySQL the STR_TO_DATE() function can do the trick in just one line!

---

Example: We want to get the date of the Tuesday of the 32^th week of the year 2013.

SELECT STR_TO_DATE('2013 32 Tuesday', '%X %V %W');

would output:

'2013-08-13'

I think this is the best and shortest solution to your problem.

The definitions of calendar week I found all said "a period of seven consecutive days starting on Sunday".

The following is MySQL specific... your mileage may vary...

DATE_ADD(MAKEDATE(year, 1), INTERVAL cw WEEK) adds the weeks from the 1st of the year which is not correct...

mysql> select DATE_ADD(MAKEDATE(2011, 1), INTERVAL 1 WEEK);
+----------------------------------------------+
| DATE_ADD(MAKEDATE(2011, 1), INTERVAL 1 WEEK) |
+----------------------------------------------+
| 2011-01-08                                   |
+----------------------------------------------+

By this definition, it is only meaningful to have the calendar week range from 1-53, and have this represent the Sunday of that week. As such, we would add 2 days to the nth Sunday of the year to get Tuesday.

The following gets the date of the first sunday of the year...

mysql> select date_add('2012-01-01', interval (8 - dayofweek('2011-01-01')) % 7 DAY);
+------------------------------------------------------------------------+
| date_add('2012-01-01', interval (8 - dayofweek('2011-01-01')) % 7 DAY) |
+------------------------------------------------------------------------+
| 2012-01-02                                                             |
+------------------------------------------------------------------------+

so this will get the date of the 10th sunday (note interval 9 week since we are already at 1)...

mysql> select date_add( date_add('2010-01-01', interval (8 - dayofweek('2010-01-01')) % 7 DAY) , interval 9 week);
+-----------------------------------------------------------------------------------------------------+
| date_add( date_add('2010-01-01', interval (8 - dayofweek('2010-01-01')) % 7 DAY) , interval 9 week) |
+-----------------------------------------------------------------------------------------------------+
| 2010-03-07                                                                                          |
+-----------------------------------------------------------------------------------------------------+

add 2 more days to get to tuesday...

mysql> select date_add( date_add( date_add('2010-01-01', interval (8 - dayofweek('2010-01-01')) % 7 DAY) , interval 9 week), interval 2 day);
+--------------------------------------------------------------------------------------------------------------------------------+
| date_add( date_add( date_add('2010-01-01', interval (8 - dayofweek('2010-01-01')) % 7 DAY) , interval 9 week), interval 2 day) |
+--------------------------------------------------------------------------------------------------------------------------------+
| 2010-03-09                                                                                                                     |
+--------------------------------------------------------------------------------------------------------------------------------+

or more generally:

select 
date_add( 
    date_add( 
        date_add('<year>-01-01', interval (8 - dayofweek('<year>-01-01')) % 7 DAY) 
        , interval <week-1> week)
    , interval <dayOfWeek> day
);

In looking at indago's answer and then doing a bunch of tests, I was getting the following week as the results.

I've made a minor adjustment, and the dates then matched:

SELECT STR_TO_DATE('2019 1 Monday', '%x %v %W') -- beginning of week

SELECT STR_TO_DATE('2019 1 Sunday', '%x %v %W') -- end of week

You can compare the results with here.