The Palm Springs Hotel was one of the city's earliest hotels, although there is little information left about it today. It was located at 265 N. Palm Canyon Drive and appears to have been around since at least the 40s. Probably before.
Through the 1950s, it was known for its Coral Tree Restaurant, which featured French-inspired food, and whose interior featured coral-colored booths with lots of tropical plants and a leaf-patterned carpet.
The Luau Restaurant looks to have been a 1940s pre-tiki establishment that was replaced by the Coral Tree. Difficult to say. Only the occasional menu turns up but judging by The Luau prices, it was probably before the Coral Tree and perhaps there are some traces of the old Luau when looking at interior shots of the Coral Tree below.
Today, this space has been entirely rebuilt and nothing of the old venue remains. It is now home to the Hyatt Palm Springs (the half which also has public self-parking).
However, you can still walk across the street from the Hyatt to where the current Tonga Hut Palm Springs offers Pop Polynesian escape to today's customers.