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In this paper, I encountered the following definition:

Definition 2 (Approximate Polynomial)

Let $U\subset \mathbb{C}$ and $\sigma\in\mathbb{N}\cup\{-\infty\}$. A function $f\colon U\to\mathbb{C}$ will be called a (right) approximate polynomial of degree $\sigma$ if the following conditions are satisfied:

  • all $u\in U$ satisfy $u+1\in U$
  • there exists a sequence of polynomials $(p_n)_{n\in\mathbb{N}}$ of fixed degree $\sigma$ such that for every $x\in U$, $\left|f(n+x) - p_n(n+x) \right| \longrightarrow 0 \quad \text{as} \;n\to+\infty$

I believed (we shall see that this might not have been the case) I followed everything up to this point, but then the paper continues as such:

This is a semi-local condition and not too restrictive; only the behavior of $f(x)$ as $\operatorname{Re}(x)\to +\infty$ matters. For example, every $f\colon\mathbb{C}\to\mathbb{C}$ with $f(x)\to 0$ as $\operatorname{Re}(x)\to+\infty$ is approximately polynomial of degree $-\infty$, and the functions $f(x)=\ln x$ and $f(x)=\sqrt x$ on $\mathbb{R}^+$ are approximately polynomial of degree $0$ (i.e. approximately constant). The class of approximate polynomials is large enough for many interesting applications.

Perhaps I am missing something quite obvious, (I haven't yet taken a formal abstract algebra or related course, nor anything more advanced) yet I fail to see how to apply the definitions above to prove the authors' claims, especially concerning finding sequences of polynomials to prove the claim.

To provide a little context/prior attempts, I first tried to tackle the natural logarithm (it grows slower than the square root so I thought it would be easier to bound). Clearly I fail understand how to choose $p_n$ correctly, as it seems to me that we can cnnot bound the logarithm with a polynomial sequence of degree $0$, even just for arguments on $(1,\infty)$ — we could simply replace the definition above with $\lim_{n \to \infty} \left|\ln(n+x) - p_n(n+x)\right|= 0$. However, I don't understand how we can pick $p_n$ of degree $0$ to satisfy this.

  1. If I understand the definition correctly we have some polynomial $p(n,x)$, where $x$ is a given complex number and $n$ is the set of integers... is this correct? I have never seen a sequence that involves more than one variable, so I'm not quite sure how they work. Will the resulting polynomial be in terms of $x$, $n$, or both? While I would assume the first, I have never before seen anything like $p_n(n+x)$, which seemingly has $n$ as both a subscript and an input variable
  2. A zeroth degree polynomial is simply a constant, yet the logarithm is strictly increasing on $(1,\infty)$, so I completely fail to see how we can choose a sequence of constants to bound it in terms of $n$... My first instinct would be to simply make the sequence $n+x$, but this appears to be a first degree polynomial, namely $1(n+x)^1$ which clearly fails the criteria. I thus guess that it is the coefficient in front of $a(n+x)^0 = a$ that needs to change, but I am not sure how to specify a correct $a(n)$ or even if this is mathematically valid.

After my attempts to prove just one of the author's claims, (on a reduced set!) it became clear to me that I lack the understanding necessary to prove the claims, and I would appreciate a simple explanation on what I currently fail to understand.

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