Give the air to

What is the meaning of the prohibition time phrase "give the air to"?

GIVE SOMEONE THE AIR
Give someone the air is American slang for reject or dismiss someone.
www.probertencyclopaedia.com/ ZGC.HTM

I am guessing that it refers to the breeze when someone passes by.

My memory is playing tricks on me. I "remember" an inquiry about "gave him the air," and I remember answering it with something like, "showed him the door, turned him away, let him have all the atmospere he needed, fired him." But I don't see any such post, so I have to conclude that I have an undiscovered brain aneurysm. SS