I honestly don't follow your first claim or how it has any bearing on this question. Perhaps you could provide a link to the actual theorem you're describing to clarify.
I misunderstood intentions. I just wanted to chime in with the name of theorem I thought you were describing. I like knowing the names of theorems, and I had a hard time finding this name when I first looked for it.
There are theorems like that for polynomials and fourier series and all sorts of other function classes too. They are just as practically relevant (or irrelevant).
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