Polynomial approximations of functions with endpoint singularities and product integration formulas

Mastroianni, Giuseppe; Monegato, Giovanni

doi:10.1090/S0025-5718-1994-1201069-2

Polynomial approximations of functions with endpoint singularities and product integration formulas
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by Giuseppe Mastroianni and Giovanni Monegato PDF

Math. Comp. 62 (1994), 725-738 Request permission

Abstract:

Several problems of mathematical physics lead to Fredholm integral equations of the second kind where the kernels are either weakly or strongly singular and the known terms are smooth. These equations have solutions which are smooth in the whole interval of integration except at the endpoints where they have mild singularities. In this paper we derive new pointwise and uniform polynomial approximation error estimates for that type of function. These estimates are then used to obtain bounds for the remainder terms of interpolatory product rules, based on the zeros of classical Jacobi orthogonal polynomials, that have been proposed for the discretization of integrals of the form \[ \int _{ - 1}^1 {k(x,y)f(x)dx,} \] appearing in the integral equations mentioned above.

References

G. A. Chandler and I. G. Graham, Product integration-collocation methods for noncompact integral operator equations, Math. Comp. 50 (1988), no. 181, 125–138. MR 917821, DOI 10.1090/S0025-5718-1988-0917821-1
Giuliana Criscuolo, Giuseppe Mastroianni, and Giovanni Monegato, Convergence properties of a class of product formulas for weakly singular integral equations, Math. Comp. 55 (1990), no. 191, 213–230. MR 1023045, DOI 10.1090/S0025-5718-1990-1023045-1
Giuliana Criscuolo, Giuseppe Mastroianni, and Péter Vértesi, Pointwise simultaneous convergence of extended Lagrange interpolation with additional knots, Math. Comp. 59 (1992), no. 200, 515–531. MR 1134723, DOI 10.1090/S0025-5718-1992-1134723-X
R. Dahlhaus, Pointwise approximation by algebraic polynomials, J. Approx. Theory 57 (1989), no. 3, 274–277. MR 999862, DOI 10.1016/0021-9045(89)90042-7
Z. Ditzian and V. Totik, Moduli of smoothness, Springer Series in Computational Mathematics, vol. 9, Springer-Verlag, New York, 1987. MR 914149, DOI 10.1007/978-1-4612-4778-4
R. V. Dudučava, Convolution integral equations with discontinuous presymbols, singular integral equations with fixed singularities, and their applications to problems in mechanics, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 60 (1979), 136 (Russian, with English summary). MR 552638
Johannes Elschner, On spline approximation for a class of noncompact integral equations, Report MATH, vol. 88, Akademie der Wissenschaften der DDR, Karl-Weierstrass-Institut für Mathematik, Berlin, 1988. With German and Russian summaries. MR 996632
Ivan G. Graham, Singularity expansions for the solutions of second kind Fredholm integral equations with weakly singular convolution kernels, J. Integral Equations 4 (1982), no. 1, 1–30. MR 640534
Giuseppe Mastroianni, Uniform convergence of derivatives of Lagrange interpolation, J. Comput. Appl. Math. 43 (1992), no. 1-2, 37–51. Orthogonal polynomials and numerical methods. MR 1193293, DOI 10.1016/0377-0427(92)90258-Y
G. Mastroianni and G. Monegato, Nyström interpolants based on the zeros of Legendre polynomials for a noncompact integral operator equation, IMA J. Numer. Anal. 14 (1994), no. 1, 81–95. MR 1254108, DOI 10.1093/imanum/14.1.81
G. Mastroianni and J. Szabados, Polynomial approximation of analytic functions with singularities, Approximation interpolation and summability (Ramat Aviv, 1990/Ramat Gan, 1990) Israel Math. Conf. Proc., vol. 4, Bar-Ilan Univ., Ramat Gan, 1991, pp. 171–181. MR 1212322
G. Monegato and V. Colombo, Product integration for the linear transport equation in slab geometry, Numer. Math. 52 (1988), no. 2, 219–240. MR 923711, DOI 10.1007/BF01398690
G. Monegato and S. Prössdorf, On the numerical treatment of an integral equation arising from a cruciform crack problem, Math. Methods Appl. Sci. 12 (1990), no. 6, 489–502. MR 1058152, DOI 10.1002/mma.1670120604
Giovanni Monegato, Product integration for one-dimensional integral equations of Fredholm type, Trends in functional analysis and approximation theory (Acquafredda di Maratea, 1989) Univ. Modena Reggio Emilia, Modena, 1991, pp. 73–86. MR 1136586
Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, DOI 10.1090/memo/0213
Paul Nevai, Mean convergence of Lagrange interpolation. III, Trans. Amer. Math. Soc. 282 (1984), no. 2, 669–698. MR 732113, DOI 10.1090/S0002-9947-1984-0732113-4
R. Piessens, Modified Clenshaw-Curtis integration and applications to numerical computation of integral transforms, Numerical integration (Halifax, N.S., 1986) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 203, Reidel, Dordrecht, 1987, pp. 35–51. MR 907110
J. Pitkäranta, On the differential properties of solutions to Fredholm equations with weakly singular kernels, J. Inst. Math. Appl. 24 (1979), no. 2, 109–119. MR 544428
Blagovest Sendov and Vasil A. Popov, The averaged moduli of smoothness, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1988. Applications in numerical methods and approximation; A Wiley-Interscience Publication. MR 995672
A. F. Timan, Theory of approximation of functions of a real variable, A Pergamon Press Book, The Macmillan Company, New York, 1963. Translated from the Russian by J. Berry; English translation edited and editorial preface by J. Cossar. MR 0192238
V. Totik, Polynomial approximation with locally geometric rate, Approximation theory (Kecskemét, 1990) Colloq. Math. Soc. János Bolyai, vol. 58, North-Holland, Amsterdam, 1991, pp. 663–671. MR 1211467
G. Vainikko and A. Pedas, The properties of solutions of weakly singular integral equations, J. Austral. Math. Soc. Ser. B 22 (1980/81), no. 4, 419–430. MR 626933, DOI 10.1017/S0334270000002769