Oh man! Once again a You Tube Good Samaritan saves the day! We've been in this apartment with these old knobs for 10 years and you just showed me how to remove them! Amazing how well made the hardware is--after 96 years of daily use they all work like they're new.
That didn't work to good for me but a hammer did a pretty decent job. The secret is to hit it from all sides of the door knob so as to not damage the door with too much force in one spot. A little "delicate force" does wonders, sometimes.