The custom is (for Ashkanazim) that a shecheyanu is not made during the three weeks. See Siman 223 Sif 3 where you see that for "celim chadashim" (new vessels) a shecheyanu is made; in practice, though, people don't make a shecheyanu on furniture nowadays. Then would it be a problem to buy new furniture for ones house? For example, a large seforim shelf or any other large piece of furniture for the house that one would probably have simcha (joy) from having.
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For Ashkenazim, it seems that an individual cannot buy furniture during the Three Weeks but a couple or a family can:
Sefardim disagree and do not allow it: