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':
- Why is the term $|\alpha + \beta| \leq m$ part of the definition?
- What does "$\alpha, \beta$ are multi-indices" mean?
- $\qquad \forall (z, \xi) \in \mathbb{R}^N \times \mathbb{R}^n$
Hopefully, an answer could address the following related questions:
- Is there a way to look at a function and say that it is polynomially bounded?
- What is an example of a function that is not polynomially bounded?
- Is a weather forecast model polynomially bounded?
Please comment if some questions would be better handled as a separate question.