I am trying to analyze the behaviour of the Associated Legendre polynomials $P_{n}^{m}$ on $[0,1]$. More specifically, I am trying to get upper bounds for $P_{n}^{m}$ on $[0,1]$. Bernstein's inequality for the Legendre polynomial is a classical result which states the following \begin{align*} |P_{n}(x)| \leq \sqrt{\frac{2}{\pi n}}\frac{1}{(1-x^2)^{1/4}}. \end{align*} I am interested in bounds of the above type for $P_{n}^{m}$'s. For a fixed $n\in \mathbb{N}$, normalizing $P_{n}^{m}$ appropriately as below, for every $|m|\leq n$ we have \begin{align*} \frac{(n-m)!}{(n+m)!} \int_{0}^{1} P_{n}^{m}(x)^2 dx = \frac{C}{2n+1}. \end{align*}
So specifically, I am interested in upper bounds on $[0,1]$ for the following functions, for a fixed $n$, as $m \in \mathbb{Z}$ varies in $[-n,n]$. \begin{align*} L_{n}^{m}(x) := \sqrt{\frac{(n-m)!}{(n+m)!}} ~P_{n}^{m} \end{align*}

I did a bit of searching and found that bounds for the above collection of normalized Associated Legendre functions are available in this and this and I state them below.

\begin{align} \sqrt{\frac{(n-m)!}{(n+m)!}}~ |P_{n}^{m} (x)| & \leq \frac{1}{2^m m!} \sqrt{\frac{(n+m)!}{(n-m)!}} (1-x^2)^{m/2} := A_{n}^{m}(x) , \tag{1}\label{1}\\ \sqrt{\frac{(n-m)!}{(n+m)!}}~ |P_{n}^{m} (x)| & \leq \frac{1}{n^{1/4}} \frac{1}{(1-x^2)^{1/8}} =: f_{n}(x).\tag{2} \label{2} \end{align}

I tried to see how good these bounds are. It seemed that $A_{n}^{m}$ is a good approximate for $L_{n}^{m}$ near 1 (which is expected as it captures the vanishing of $L_{n}^{m}$ at 1), but elsewhere it is not good.

And the bound $f_n$ appears to be just an upper bound which does not necessarily capture any feature of $L_{n}^{m}$.

Hence my question is: Are some other better bounds (than the ones in \ref{1} and \ref{2}) known for $L_{n}^{m}$'s?



1 Answer 1


Please have a look at the following paper


In particular, Eq. (6) to get the answer to your question.

Lohöfer, G., Inequalities for the associated Legendre functions, J. Approximation Theory 95, No. 2, 178-193 (1998). ZBL0923.33002.


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