A trademark search would help as a start but is not the whole answer. Smith as a common name has no protection against the other Smiths in the world. Smith Publishing is not distinctive in itself. If it is registered, there could be a serious problem for you. If it is not the whole fact picure must be analized.
Trade names are less enforceable but can be enforced against unfair competition.
There are solutions that can be found. Don't look for trouble. It costs less to be safe now than sorry later. I believe the product line to be different. The whole fact picture must be known to be able to answer the question.