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Mathematics of Computation

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Asymptotic expansions of Legendre series coefficients for functions with interior and endpoint singularities
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by Avram Sidi PDF
Math. Comp. 80 (2011), 1663-1684 Request permission

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 $x=d$.
References
  • Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. 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, DOI 10.1017/CBO9781107325937
  • 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 10.1016/0096-3003(81)90028-X
  • F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
  • Daniel Shanks, Non-linear transformations of divergent and slowly convergent sequences, J. Math. and Phys. 34 (1955), 1–42. MR 68901, DOI 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, DOI 10.1017/CBO9780511546815
  • 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 Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. 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
  • 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.
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2785473