Did Denethor see Frodo captured in the Palantir?

[Galadriel55]

I believe that Denethor could and did resist Sauron for a good while. However, his mental defense couldn't have lasted too long. I don't think he gave away too much information over the years of using the Palantir. He was weakened by the end - but that's because of all his previous hard-won victories. Aragorn isn't the only strong-willed guy around.

Originally Posted by FerniesApple

I have a theory that Sauron never contemplated that the Ring was in the hands of the Hobbits until he 'saw' Pippin in Minas Tirith through the Palantir Denethor was using.

What happened to "Shire... Baggins..."? Sauron knew a hobbit had the Ring for a long time. Furthermore, how is this relevant? Sauron saw Pippin when the said hobbit looked into Saruman's Palantir, much before he came to Minas Tirith. And either way, how does the Ring itself tie in?