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Let $P_n:[-1,1]\rightarrow\mathbb{R}$ be the $n$-th Legendre Polynomial. and, let $$a_n:=\int_{-1}^1 f(t) P_n(t)\, dt$$ for some $f:[-1,1]\rightarrow\mathbb{R}$.

Are there any good references on the decay rate of $\vert a_n \vert$? I am not familiar with this kind of problem, but I guess there must be a lot of methods.

From the following similar mathoverflow question: Reference for the exponential decay of Legendre coefficients, I found one paper. Also, I could find a book "Spherical Harmonics and Approximations on the Unit Sphere" by Atkinson. After skimming those references, it seems that showing smoothness of $f$ is one method. But the application in my mind is the case when $f=\arccos^2(t)$, which is not smooth enough.

So, I'm wondering are there any other references explaniing various methods to compute decay rate of $\vert a_n \vert$. Especially, if there are some techniques one can use for non-smooth $f$, I really want to know.

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  • $\begingroup$ You would have to assume some kind of smoothness. Otherwise the decay could be arbitrarily slow (e.g. $f(t)=\sum_{n>0}2^{-n}P_{n!!!!!}$). $\endgroup$
    – Anthony Quas
    Commented Jan 10, 2020 at 3:40
  • $\begingroup$ You might find an article by Guillemot-Tessier in the Ann. Scuola Norm. Sup. Pisa, 25 (1971) 519-573 interesting (available online). She is interested in the extreme cases (smooth test functions, with rapidly decreasing coefficients, and distributions, with slowly increasing ones) but gives estimates which could be useful for other situations. $\endgroup$
    – user131781
    Commented Jan 10, 2020 at 4:04
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    $\begingroup$ @Anthony Quas Thank you. So, $f$ should be smooth on whole $[-1,1]$? If $f$ is smooth only interior $(-1,1)$ Is there any way to control the decay rate? $\endgroup$
    – S.Lim
    Commented Jan 10, 2020 at 4:05
  • $\begingroup$ @user131781 Thank you. I will look into it. $\endgroup$
    – S.Lim
    Commented Jan 10, 2020 at 4:06

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