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Why do we require $\delta_a \in \mathbb{Z}$ in the definition of an approximation space?
I’m currently taking a course in approximation theory. In most of the material we need $\delta_a \in \mathbb{Z}$. In some cases, $\mathbb{Z}$ is relaxed to $\mathbb{Q}$ and is omitted. (see section 1.9 here)
Why do we need $\delta_a \in \mathbb{Z}$?
A:
I don’t think your book is using “approximation space” in the sense of Nevanlinna or Nussbaum (as say was customary at the time these books were written), but rather in the sense of Polya. The reason for this more restrictive definition is that it makes the theory simpler. Indeed, $\delta_a = 0$ is no longer an admissible choice.
From the classical theory of Nevanlinna (1952), which was used in the early books on the subject, to the current theory of Nussbaum, which allows one to take $\delta_a$ arbitrary and even $0$, we need to pass a “backward step”. Indeed, $0$ was the acceptable thing to do at the time when these two very important initial books were written.
Nowadays, one needs to pass another “backward step” to allow $\delta_a$ to be an integer.
It’s a funny history that, in the “golden age” of approximation theory, the finite union of the open intervals $(a,\delta_a)$ with $a$ rational, $0
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