The 37-year-old was found with serious injuries in Houston Street, close to Wilson Street on Sunday, 14 January.
He was taken to the Queen Elizabeth University Hospital, however he died the following week in the early hours of Saturday, 20 January.
Following his death, officers launched a murder inquiry and are continuing to work to ascertain the circumstances of what happened.
A 26-year-old man and a 22-year-old women were arrested in connection with the death and later released pending further enquiries.
Detective Chief Inspector Martin Fergus from Police Scotland’s Major Investigation Team (West) said:
“It is coming up to 10 weeks since Paul Mathieson was found seriously injured in Renfrew and we still need anyone with information to come forward to assist us in piecing together the circumstances of what happened to him and provide answers for his family.
“We have made two arrests in connection with the death, however both people have been released pending further enquiries and this is still very much an ongoing police investigation.
“I would ask anyone who was in the Renfrew area in the early hours of Sunday 14 January, who may have noticed anything at all suspicious to please get in touch.
“I would also appeal directly to anyone who has information about what happened to Mr Mathieson, or who is responsible, to look into your conscience and come forward. I am confident someone has information that could lead to us finding answers for his family, so please do the right thing and tell us what you know.
“Anyone with information is asked to contact officers from the Major Investigation Team (West) via 101, quoting incident number 0309 of 14 January 2018. Alternatively you can call CRIMESTOPPERS on 0800 555 111 where details can be given in confidence.”