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

Author:
Avram Sidi

Journal:
Math. Comp. **80** (2011), 1663-1684

MSC (2000):
Primary 40A05, 40A10, 41A58, 41A60, 42C10

DOI:
https://doi.org/10.1090/S0025-5718-2010-02454-8

Published electronically:
December 30, 2010

MathSciNet review:
2785473

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Abstract | References | Similar Articles | Additional Information

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 $x=d$.

- Milton Abramowitz and Irene A. Stegun,
*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642** - George E. Andrews, Richard Askey, and Ranjan Roy,
*Special functions*, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR**1688958** - N. Bleistein and R.A. Handelsman,
*Asymptotic expansions of integrals*, Holt, Rinehart and Winston, New York, 1975. - T.T. Cīrulis,
*Asymptotic representation of the Fourier coefficients of functions in series of Legendre polynomials (Russian)*, Latviĭsk. Mat. Ežegodnik Vyp.**19**(1976), 47–62, 243–244. - M. K. Jain and M. M. Chawla,
*The estmation of the coefficients in the Legendre series expansion of a function*, J. Mathematical and Physical Sci.**1**(1967), 247–260. MR**244695** - David Levin and Avram Sidi,
*Two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series*, Appl. Math. Comput.**9**(1981), no. 3, 175–215. MR**650681**, DOI https://doi.org/10.1016/0096-3003%2881%2990028-X - F. W. J. Olver,
*Asymptotics and special functions*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR**0435697** - Daniel Shanks,
*Non-linear transformations of divergent and slowly convergent sequences*, J. Math. and Phys.**34**(1955), 1–42. MR**68901**, DOI https://doi.org/10.1002/sapm19553411 - A. Sidi,
*Some properties of a generalization of the Richardson extrapolation process*, J. Inst. Math. Appl.**24**(1979), no. 3, 327–346. MR**550478** - Avram Sidi,
*Acceleration of convergence of (generalized) Fourier series by the $d$-transformation*, Ann. Numer. Math.**2**(1995), no. 1-4, 381–406. Special functions (Torino, 1993). MR**1343544** - Avram Sidi,
*Practical extrapolation methods*, Cambridge Monographs on Applied and Computational Mathematics, vol. 10, Cambridge University Press, Cambridge, 2003. Theory and applications. MR**1994507** - Avram Sidi,
*Asymptotic expansions of Legendre series coefficients for functions with endpoint singularities*, Asymptot. Anal.**65**(2009), no. 3-4, 175–190. MR**2574341** - ---,
*Asymptotic analysis of a generalized Richardson extrapolation process on linear sequences*, Math. Comp.**79**(2010), 1681–1695. - ---,
*A simple approach to asymptotic expansions for Fourier integrals of singular functions*, Appl. Math. Comput.**216**(2010), 3378–3385. MR**2653157** - Gábor Szegő,
*Orthogonal polynomials*, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR**0372517**

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Additional Information

**Avram Sidi**

Affiliation:
Computer Science Department, Technion, Israel Institute of Technology, Haifa 32000, Israel

Email:
asidi@cs.technion.ac.il

Keywords:
Legendre polynomials,
Legendre series,
interior singularities,
endpoint singularities,
asymptotic expansions.

Received by editor(s):
March 17, 2010

Received by editor(s) in revised form:
May 25, 2010

Published electronically:
December 30, 2010

Additional Notes:
This research was supported in part by the United States–Israel Binational Science Foundation grant no. 2008399.

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.