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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.