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

Abstract

Let ∑ n = 0 ∞ e n [ f ] P n ( x ) \sum ^\infty _{n=0} e_n[f] P_n(x) be the Legendre expansion of a function f ( x ) f(x) on ( − 1 , 1 ) (-1,1) . In an earlier work [A. Sidi, Asymptot. Anal. , 65 (2009), pp. 175–190], we derived asymptotic expansions as n → ∞ n\to \infty for e n [ f ] e_n[f] , assuming that f ∈ C ∞ ( − 1 , 1 ) f\in C^\infty (-1,1) , but may have arbitrary algebraic-logarithmic singularities at one or both endpoints x = ± 1 x=\pm 1 . In the present work, we extend this study to functions f ( x ) f(x) that are infinitely differentiable on [ 0 , 1 ] [0,1] , except at finitely many points x 1 x_1 , …, x m x_m in ( − 1 , 1 ) (-1,1) and possibly at one or both of the endpoints x 0 = 1 x_0=1 and x m + 1 = − 1 x_{m+1}=-1 , where they may have arbitrary algebraic singularities, including finite jump discontinuities. Specifically, we assume that, for each r r , f ( x ) f(x) has asymptotic expansions of the form \[ f ( x ) ∼ ∑ s = 0 ∞ W r s ( ± ) | x − x r | δ r s ( ± ) as  x → x r ± , f(x) \sim \sum ^\infty _{s=0} W^{(\pm )}_{rs} |x-x_r|^{\delta ^{(\pm )}_{rs}} \quad \text {as $x\to x_r\pm $}, \] where W r s ( ± ) W^{(\pm )}_{rs} and δ r s ( ± ) \delta ^{(\pm )}_{rs} are, in general, complex and ℜ δ r s ( ± ) > − 1 \Re \delta ^{(\pm )}_{rs}>-1 . We derive the full asymptotic expansion of e n [ f ] e_n[f] as n → ∞ n\to \infty for this very general behavior of f ( x ) f(x) . In the special case where δ r s ( ± ) = σ r ( ± ) + s \delta ^{(\pm )}_{rs}=\sigma ^{(\pm )}_{r}+s , 1 ≤ r ≤ m 1\leq r\leq m , and δ 0 s ( − ) = α + s \delta ^{(-)}_{0s}=\alpha +s and δ m + 1 , s ( + ) = β + s \delta ^{(+)}_{m+1,s}=\beta +s , this expansion reduces to e n [ f ] ∼ ∑ r = 1 m { e i n ^ θ r a m p ; [ ∑ s = 0 ∞ a r s ( + ) n ^ σ r ( + ) + s + 1 / 2 + ∑ s = 0 ∞ a r s ( − ) n ^ σ r ( − ) + s + 1 / 2 ] + e − i n ^ θ r a m p ; [ ∑ s = 0 ∞ a ^ r s ( + ) n ^ σ r ( + ) + s + 1 / 2 + ∑ s = 0 ∞ a ^ r s ( − ) n ^ σ r ( − ) + s + 1 / 2 ] } + ∑ s = 0 α ∉ Z + ∞ a m p ; A s n ^ 2 ( α + s + 1 / 2 ) + ( − 1 ) n ∑ s = 0 β ∉ Z + ∞ B s n ^ 2 ( β + s + 1 / 2 ) as  n → ∞ . \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 ot \in \mathbb {Z}^+}} &\frac {A_s}{\widehat {n}^{2(\alpha +s+1/2)}} +(-1)^n \sum ^\infty _{\substack {s=0 \\ \beta ot \in \mathbb {Z}^+}} \frac {B_s} {\widehat {n}^{2(\beta +s+1/2)}}\quad \text {as $n\to \infty $.} \end{align*} where θ r = cos − 1 ⁡ x r \theta _r=\cos ^{-1}x_r , n ^ = n + 1 / 2 \widehat {n}=n+1/2 , Z + = { 0 , 1 , 2 , … } \mathbb {Z}^+=\{0,1,2,\ldots \} , and a r s ( ± ) a^{(\pm )}_{rs} , a ^ r s ( ± ) \widehat {a}^{(\pm )}_{rs} , A s A_s , and B s B_s are constants independent of n n . In the course of this study, we also derive a full asymptotic expansion as n → ∞ n\to \infty for integrals of the form ∫ c d f ( x ) P n ( x ) d x \int ^d_cf(x)P_n(x)\,dx where [ c , d ] ∈ ( − 1 , 1 ) [c,d]\in (-1,1) and f ∈ C ∞ [ c , d ] f\in C^\infty [c,d] or f ∈ C ∞ ( c , d ) f\in C^\infty (c,d) but may have arbitrary algebraic singularities at x = c x=c and/or x = d x=d .