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Let $P_n$ be the Legendre polynomial of degree $n$. Could you suggest me some references to bound the polynomials on the complex plane (near the real line in particular) ? More specifically, I need to prove that $$\sum_{i=1}^{n}\sqrt i |P_i(z)|^2 >> n^{a}$$ for some $a>0$ and for all $z$ with $\Re(z)\in [-1+b, 1-b], |\Im(z)|\le 1/n$, for a given $b>0$. I also need upper bounds on $|P_i(z)|$, $|P'_i(z)|$, and $|P''_i(z)|$ for such $z$ as above. Any suggestion is greatly appreciated.

I know some bounds on the real line, such as the Bernstein and Markov inequalities, which say that $P_i(x)$ is of order $i^{-1/2}$ for $x \in [-1+b, 1-b]$. But I have no clue how to work with $x$ in the complex plane.

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  • $\begingroup$ your $P_n$ is monic or orthonormal ? $\endgroup$
    – user111
    Sep 6, 2017 at 12:24
  • $\begingroup$ It's the standard Legendre polynomials as in here en.wikipedia.org/wiki/Legendre_polynomials. They are orthogonal but neither monic nor orthonormal. $\endgroup$
    – Doanh Doanh
    Sep 6, 2017 at 12:35

1 Answer 1

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Setting $Q_{n}(z)=(n+1/2)^{1/2}P_{n}(z)$ for the orthonormalized Legendre polynomial, and $$\kappa_{n}(z)=\sum_{j=0}^{n}|Q_{j}(z)|^{2},$$ for the inverse of the Christoffel function, it is known that $$\kappa_{n}^{\frac{1}{2n}}(z)\to\left|z+\sqrt{z^{2}-1}\right|,$$ uniformly on $\mathbb{C}$ as $n\to\infty$, so that $\kappa_{n}(z)$ grows geometrically fast outside of $[-1,1]$. For $x\in[-1,1]$, a deeper result is needed, which says that $$\lim_{n\to\infty}\frac{\kappa_{n}(x)}{n}=\frac{1}{\pi\sqrt{1-x^{2}}},$$ see, for instance, Chapter V.6. of the book

G. Freud, Orthogonal Polynomials, Pergamon Press, New York, 1971.

Hence, for $x\in[-1,1]$, $$\lim_{n\to\infty}\frac1n\sum_{j=0}^{n}\left(j+\frac12\right)|P_{j}(x)|^{2}=\frac{1}{\pi\sqrt{1-x^{2}}},$$ from which one may derive, together with Abel's summation formula, that $\sum_{j=0}^{n}\sqrt{j}|P_{j}(x)|^{2}$ behaves, for $x\in(-1,1)$ and $n$ large, like $C_{x}\sqrt{n}$ where $C_{x}$ is some positive, bounded below, constant depending on $x$.

Hence your estimate in a neighborhood of $[-1,1]$ holds true with the exponent $a=1/2$ which is optimal.

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  • $\begingroup$ For a fixed point x, then using the limits that you mentioned, we get the bound $\kappa_n(x) \gg n^a$. I need a bound that holds for every x, i.e. the implied constant does not depend on x. Other than the limits, do you know any more precise estimates? Btw, do you know any bound like $|Q'(z)| << n$ for $z$ in the question? Thanks a bunch! $\endgroup$
    – Doanh Doanh
    Sep 7, 2017 at 20:51
  • $\begingroup$ yes, the constants $C_x$, $x\in(-1,1)$, are bounded below by some positive constant $C$ independent of $x$. For a bound on the derivative, see e.g. mathoverflow.net/questions/151978/…. $\endgroup$
    – user111
    Sep 8, 2017 at 5:23
  • $\begingroup$ But it doesn't work for x not real, does it? $\endgroup$
    – Doanh Doanh
    Sep 9, 2017 at 12:15
  • $\begingroup$ Does $z+\sqrt{z^2-1}$ vanish outside of $[-1,1]$ ? $\endgroup$
    – user111
    Sep 9, 2017 at 14:07
  • $\begingroup$ Thanks for the replies. I get the part about $\kappa$. My question was that the Bernstein inequality in mathoverflow.net/questions/151978/… only seems to work for real x. $\endgroup$
    – Doanh Doanh
    Sep 9, 2017 at 19:26

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