Schönberg or Schoenberg???

I am doing some work at the Arnold Schönberg Center while I'm in Vienna, and I've often wondered why they spell his name with an "ö" instead of "oe." The family asked that it be spelled with the umlaut even though Arnold Schoenberg officially changed the spelling after emigrating to the United States, and the current family members on the board at the Center spell it "oe." I guess that's not too important. Schoenberg was one of many Jews who were forced to flee Austria in the era preceding World War II. One very positive thing his family did do, from what I have been told, is to make Austria publicly apologize for the atrocities committed against Jews in Austria. This was the condition that they gave in order to have the Schönberg Center in Austria along with the materials they donated to the Center. The facility is fairly new and has an excellent collection of music reference books in general, not just those dealing with Schoenberg, and the staff are very friendly and helpful.