Optimal decay rates on the asymptotics of orthogonal polynomial expansions for functions of limited regularities
Tóm tắt
Từ khóa
Tài liệu tham khảo
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C. (1964)
Antonov, V.A., Holsevnikov, K.V., Shaidulin, V.S.: Estimating the Derivative of the Legendre Polynomial, Vestnik St. Petersburg University. Mathematics 43(4), 191–197 (2010)
Bernstein, S.: Sur l’order de la meilleure approximation des fonctions continues par des polynomes de degré donné. Mem. Acad. R. Belg. 4, 1–103 (1912)
Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover Publications, New York (2000)
Boyd, J.P., Petschek, R.: The relationships between Chebyshev, Legendre and Jacobi polynomials: the generic superiority of Chebyshev polynomials and three important exceptions. J. Sci. Comput. 59, 1–27 (2014)
Dahlquist, G., Björck, A.: Numerical Methods in Scientific Computing. SIAM, Philadelphia (2007)
Darboux, G.: Mémoire sur l’approximation des fonctions de très-grands nombres et sur une classe étendue de développements en série. J. Math. Pure Appl. 4, 5–56 (1978)
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, New York (1984)
Don, W.S., Gottlieb, D.: The Chebyshev–Legendre method: implementing Legendre methods on Chebyshev points. SIAM J. Numer. Anal. 31, 1519–1534 (1994)
Driscoll, T.A., Hale, N., Trefethen, L.N. (eds.): Chebfun User’s Guide. Pafnuty Publications, Oxford (2014)
Durand, L.: Nichelson-type integrals for products of Gegenbauer functions and related topics. In: Askey, R.A. (ed.) Theory and Application of Special Functions, pp. 353–374. Academic Press, New York (1975)
Elliott, D.: The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function. Math. Comput. 18, 274–284 (1964)
Erdélyi, A.: Asymptotic representations of Fourier integrals and the method of stationary phase. J. Soc. Ind. Appl. Math. 3, 17–27 (1955)
Erdélyi, A.: Asymptotic expansions of Fourier integrals involving logarithmic singularities. J. Soc. Ind. Appl. Math. 4, 38–47 (1956)
Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1996)
Fox, L., Parker, I.B.: Chebyshev Polynomials in Numerical Analysis. Oxford University Press, London (1968)
Gautschi, W.: Orthogonal Polynomials Computation and Approximation, pp. 1170–1186. Oxford University Press, London (2004)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products, 7th edn. Academic Press, New York (2007)
Haagerup, U., Schlichtkrull, H.: Inequalities for Jacobi polynomials. Ramanujan J. 33, 227–246 (2014)
Hale, N., Townsend, A.: Fast and accurate computation of Gauss–Legendre and Gauss–Jacobi quadrature nodes and weights. SIAM J. Sci. Comput. 35, A652–A674 (2013)
Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007)
Lanczos, C.: Tables of Chebyshev Polynomials $$S_n(x)$$ and $$C_n(x)$$. National Bureau of Standards Applied Mathematics, vol. 9. U.S. Government Printing Office, Washington D.C (1952)
Liu, W., Wang, L., Li, H.: Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces. Math. Comput. 88, 2857–2895 (2019)
Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. CRC Press, New York (2003)
Miller, G.F.: On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal. 3(3), 390–409 (1966)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)
Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. Springer, Berlin (2011)
Szegö, G.: Orthogonal Polynomial. Academic Mathematical Society, Providence (1939)
Trefethen, L.N.: Approximation Theory and Approximation Practice. SIAM, Philadelphia (2013)
Tuan, P.D., Elliott, D.: Coefficients in series expansions for certain classes of functions. Math. Comput. 26, 213–232 (1972)
Wang, H.: Convergence rate and acceleration of Clenshaw–Curtis quadrature for functions with endpoint singularities. J. Comput. Appl. Math. 333(2018), 87–98 (2014). (arXiv 1401.0638)
Wang, H., Xiang, S.: On the convergence rates of Legendre approximation. Math. Comput. 81, 861–877 (2012)
Wang, H.: On the optimal estimates and comparison of Gegenbauer expansion coefficients. SIAM J. Numer. Anal. 34, 1557–1580 (2016)
Wang, H.: A new and sharper bound for Legendre expansion of differentiable functions. Appl. Math. Lett. 85, 95–102 (2018)
Watson, G.N.: A Tretise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)
Xiang, S.: Numerical analysis of a fast integration methods for highly oscillatory functions. BIT Numer. Anal. 47, 469–482 (2007)
Xiang, S.: On error bounds for orthogonal polynomial expansions and Gauss-Type Quadrature. SIAM J. Numer. Anal. 50, 1240–1263 (2012)
Xiang, S.: On interpolation approximation: convergence rates for polynomial interpolation for function of limited regularity. SIAM J. Numer. Anal. 54, 2081–2113 (2016)
Xiang, S.: On the optimal convergence rates of Chebyshev interpolations for functions of limited regularity. Appl. Math. Lett. 84, 1–7 (2018)
Xiang, S.: On van der Corput-type lemmas for Bessel and Airy transforms and applications. J. Comput. Appl. Math. 351, 179–185 (2019)
Xiang, S., Brunner, H.: Efficient methods for Volterra integral equations with highly oscillatory Bessel kernels. BIT Numer. Anal. 53, 4241–4263 (2013)
Xie, Z., Wang, L., Zhao, X.: On exponential convergence of Gegenbauer interpolation and spectral differentiation. Math. Comput. 82, 1017–1036 (2013)
Zhao, X., Wang, L., Xie, Z.: Sharp error bounds for Jacobi expansions and Gegenbauer–Gauss quadrature of analytic functions. SIAM J. Numer. Anal. 251, 1443–1469 (2012)