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I am currently reading through some lecture notes on Fourier Transform and Distributions on my own, and came upon the notion of a polynomially bounded function. I am not sure I understand this concept, so I would like to ask

What does it mean for a function $G(z,\xi)$ (where $z \in \mathbb{R}^N$ and $\xi \in \mathbb{R}^n$) to be polynomially bounded?

My guess is that this means there exists an integer $m$ and complex coefficients $c_{\alpha,\beta}$ such that the polynomial

\begin{equation} P(z,\xi) = \sum_{|\alpha + \beta| \leq m} c_{\alpha,\beta}\, z^\alpha \xi^\beta \qquad (\alpha,\ \beta \text{ are multi-indices}) \end{equation} satisfies \begin{equation} |G(z,\xi)| \leq \, |P(z,\xi)| \qquad \forall \, (z, \xi) \in \mathbb{R}^N \times \mathbb{R}^n. \end{equation}

Do the expressions above make sense, and is my interpretation of polynomial boundedness correct?

Many thanks and Regards!

Specific points for posted bounty:

This question was given as a reference to a question on the Delta Method on stats.SE.

Could this be explained to a high school algebra student? If not, to a college student who has taken probability and linear algebra? If so, that would be most helpful.

Specifically, it would help to have explanations of the following terms, and why they are required to define 'polynomially bounded':

  1. Why is the term $|\alpha + \beta| \leq m$ part of the definition?
  2. What does "$\alpha, \beta$ are multi-indices" mean?
  3. $\qquad \forall (z, \xi) \in \mathbb{R}^N \times \mathbb{R}^n$

Hopefully, an answer could address the following related questions:

  1. Is there a way to look at a function and say that it is polynomially bounded?
  2. What is an example of a function that is not polynomially bounded?
  3. Is a weather forecast model polynomially bounded?

Please comment if some questions would be better handled as a separate question.

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Yes, that's correct. Moreover, you can say that there exists $m$ such that outside a ball $B_r(0)$ around zero in $\mathbb{R}^{N + n}$ of sufficiently large radius $r > 0$, $|G(z,\xi)|\leq |(z,\xi)|^m$, where $|\cdot|$ denotes an arbitrary norm in $\mathbb{R}^{N+n}$ (i.e. a norm chosen arbitrarily and fixed prior to choosing $m$ and $r$). – user12918723509187 Feb 14 '12 at 20:16
that's great, thanks for your help! – harlekin Feb 14 '12 at 20:55

To address the explanations:

  1. A multi-index $\alpha=(\alpha_1,...,\alpha_n)$ is an element of $R^n$ with $\alpha_i \in N$ (non-negative integers). $|\alpha| = \sum_i \alpha_i$ is the order of a multi-index. And for $z=(z_1,...,z_n)\in R^n$, the term $z^\alpha$ means the product $z_1^{\alpha_1}...z_n^{\alpha_n}$.
  2. The term $\alpha + \beta$ only makes sense, if $n = N$ which is not guaranteed. So $|\alpha|+|\beta| <= m$ should be used meaning the sum over all multi-indices with the sum of their orders is less or equal $m$.
  3. $\forall$ means for all.

So basically, this is the formal definition of a polynomial in several variables of order at most $m$.

To address the questions:

  1. Polynomially bounded is basically saying that the function is $\mathcal{O}(|x|^m)$. Can you say this by looking at a function.
  2. The exponential function is not polynomially bounded, as $\frac{e^x}{p(x)} \to \infty$ as $x\to\infty$ for any polynomial $p(x)$.
  3. No idea about the weather forecast...

For me this sounds like: If a function is polynomially bounded, its Taylor remainder is somehow getting small, if the degree is higher than the degree of the polynomial.

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