Robbie Morrison, 31, was attacked in Dumbryden Gardens in Wester Hailes at around 4.30pm last Friday.
He was taken to Edinburgh Royal Infirmary but died on Monday.
Officers have been carrying out high-visibility patrols and speaking to local residents in an attempt to identify the attacker.
Detective Chief Inspector Gary Cunningham said: "A week has passed now since this horrific attack on Mr Morrison, and while our officers are making progress with the investigation, we would urge anyone with any information to come forward. It is clear from our ongoing inquiries that the community holds vital information that would assist us.
"Seven days on, our officers will today have a very visible presence in Wester Hailes and will be asking members of the public if they saw anything suspicious or individuals in Dumbryden Gardens last Friday afternoon or, indeed, if they have heard something which may prove key to our investigation."
Chief Inspector Mark Rennie added: "Since Robbie Morrison was attacked and subsequently died, we have been keen to reassure residents in Wester Hailes that this appears to be an isolated incident.
"Our officers are doing everything they can to trace the person who carried out this murder and we will continue to carry out patrols in the area."