Asymptotic expansions of Legendre series coefficients for functions with interior and endpoint singularities

Abstract:

Let $\sum ^\infty _{n=0} e_n[f] P_n(x)$ be the Legendre expansion of a function $f(x)$ on $(-1,1)$. In an earlier work [A. Sidi, Asymptot. Anal., 65 (2009), pp. 175–190], we derived asymptotic expansions as $n\to \infty$ for $e_n[f]$, assuming that $f\in C^\infty (-1,1)$, but may have arbitrary algebraic-logarithmic singularities at one or both endpoints $x=\pm 1$. In the present work, we extend this study to functions $f(x)$ that are infinitely differentiable on $[0,1]$, except at finitely many points $x_1$, …, $x_m$ in $(-1,1)$ and possibly at one or both of the endpoints $x_0=1$ and $x_{m+1}=-1$, where they may have arbitrary algebraic singularities, including finite jump discontinuities. Specifically, we assume that, for each $r$, $f(x)$ has asymptotic expansions of the form \[ f(x) \sim \sum ^\infty _{s=0} W^{(\pm )}_{rs} |x-x_r|^{\delta ^{(\pm )}_{rs}} \quad \text {as $x\to x_r\pm $}, \] where $W^{(\pm )}_{rs}$ and $\delta ^{(\pm )}_{rs}$ are, in general, complex and $\Re \delta ^{(\pm )}_{rs}>-1$. We derive the full asymptotic expansion of $e_n[f]$ as $n\to \infty$ for this very general behavior of $f(x)$. In the special case where $\delta ^{(\pm )}_{rs}=\sigma ^{(\pm )}_{r}+s$, $1\leq r\leq m$, and $\delta ^{(-)}_{0s}=\alpha +s$ and $\delta ^{(+)}_{m+1,s}=\beta +s$, this expansion reduces to \begin{align*}e_n[f]\sim \sum ^{m}_{r=1} \bigg \{ e^{\mathrm {i}\widehat {n}\theta _r}&\bigg [\sum ^\infty _{s=0} \frac {a^{(+)}_{rs}} {\widehat {n}^{\sigma ^{(+)}_{r}+s+1/2}}+ \sum ^\infty _{s=0} \frac {a^{(-)}_{rs}} {\widehat {n}^{\sigma ^{(-)}_{r}+s+1/2}}\bigg ] \\ +e^{-\mathrm {i}\widehat {n}\theta _r}&\bigg [\sum ^\infty _{s=0} \frac {\widehat {a}^{(+)}_{rs}} {\widehat {n}^{\sigma ^{(+)}_{r}+s+1/2}}+ \sum ^\infty _{s=0} \frac {\widehat {a}^{(-)}_{rs}} {\widehat {n}^{\sigma ^{(-)}_{r}+s+1/2}}\bigg ] \bigg \}\\ +\sum ^\infty _{\substack {s=0 \\ \alpha \not \in \mathbb {Z}^+}} &\frac {A_s}{\widehat {n}^{2(\alpha +s+1/2)}} +(-1)^n \sum ^\infty _{\substack {s=0 \\ \beta \not \in \mathbb {Z}^+}} \frac {B_s} {\widehat {n}^{2(\beta +s+1/2)}}\quad \text {as $n\to \infty $.} \end{align*} where $\theta _r=\cos ^{-1}x_r$, $\widehat {n}=n+1/2$, $\mathbb {Z}^+=\{0,1,2,\ldots \}$, and $a^{(\pm )}_{rs}$, $\widehat {a}^{(\pm )}_{rs}$, $A_s$, and $B_s$ are constants independent of $n$. In the course of this study, we also derive a full asymptotic expansion as $n\to \infty$ for integrals of the form $\int ^d_cf(x)P_n(x)\,dx$ where $[c,d]\in (-1,1)$ and $f\in C^\infty [c,d]$ or $f\in C^\infty (c,d)$ but may have arbitrary algebraic singularities at $x=c$ and/or $x=d$.