On the convergence of an interpolatory product rule for evaluating Cauchy principal value integrals

Authors:
Giuliana Criscuolo and Giuseppe Mastroianni

Journal:
Math. Comp. **48** (1987), 725-735

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878702-4

MathSciNet review:
878702

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The authors give convergence theorems for interpolatory product rules for evaluating Cauchy singular integrals and obtain asymptotic estimates of the remainder. Some results, previously established by other authors, are generalized and improved.

**[1]**N. K. Bari and S. B. Stečkin,*Best approximations and differential properties of two conjugate functions*, Trudy Moskov. Mat. Obšč.**5**(1956), 483–522 (Russian). MR**0080797****[2]**Giuliana Criscuolo and Giuseppe Mastroianni,*The convergence of some quadrature formulas for Cauchy principal value integrals*, Anal. Numér. Théor. Approx.**14**(1985), no. 2, 109–116 (Romanian, with English summary). MR**850730****[3]**C. Dagnino and A. Palamara Orsi,*On the evaluation of certain two-dimensional singular integrals with Cauchy kernels*, Numer. Math.**46**(1985), no. 1, 121–130. MR**777828**, https://doi.org/10.1007/BF01400259**[4]**David Elliott and D. F. Paget,*On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals*, Numer. Math.**23**(1975), 311–319. MR**0380215**, https://doi.org/10.1007/BF01438258**[5]**David Elliott and D. F. Paget,*“On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals” (Numer. Math. 23 (1975), 311–319): an addendum*, Numer. Math.**25**(1975/76), no. 3, 287–289. MR**0411131**, https://doi.org/10.1007/BF01399417**[6]**G. Freud,*On expansions in orthogonal polynomials*, Studia Sci. Math. Hungar.**6**(1971), 367–374. MR**0294981****[7]**N. I. Ioakimidis,*Further convergence results for two quadrature rules for Cauchy type principal value integrals*, Apl. Mat.**27**(1982), no. 6, 457–466 (English, with Czech summary). With a loose Russian summary. MR**678115****[8]**Dunham Jackson,*The theory of approximation*, American Mathematical Society Colloquium Publications, vol. 11, American Mathematical Society, Providence, RI, 1994. Reprint of the 1930 original. MR**1451140****[9]**G. Monegato,*The numerical evaluation of one-dimensional Cauchy principal value integrals*, Computing**29**(1982), no. 4, 337–354 (English, with German summary). MR**684742**, https://doi.org/10.1007/BF02246760**[10]**Paul G. Nevai,*Orthogonal polynomials*, Mem. Amer. Math. Soc.**18**(1979), no. 213, v+185. MR**519926**, https://doi.org/10.1090/memo/0213**[11]**Paul Nevai,*Mean convergence of Lagrange interpolation. III*, Trans. Amer. Math. Soc.**282**(1984), no. 2, 669–698. MR**732113**, https://doi.org/10.1090/S0002-9947-1984-0732113-4**[12]**M. A. Sheshko, "On the convergence of quadrature processes for a singular integral,"*Soviet Math.*(*Iz. VUZ.*), v. 20, no. 12, 1976, pp. 86-94.**[13]**M. A. Šeško and T. S. Jakimenko,*The convergence of a quadrature process for a singular integral with power-logarithmic singularity*, Izv. Vyssh. Uchebn. Zaved. Mat.**1**(1980), 82–84 (Russian). MR**567606****[14]**William E. Smith and Ian H. Sloan,*Product-integration rules based on the zeros of Jacobi polynomials*, SIAM J. Numer. Anal.**17**(1980), no. 1, 1–13. MR**559455**, https://doi.org/10.1137/0717001**[15]**Ian H. Sloan and William E. Smith,*Properties of interpolatory product integration rules*, SIAM J. Numer. Anal.**19**(1982), no. 2, 427–442. MR**650061**, https://doi.org/10.1137/0719027**[16]**G. Szegö,*Orthogonal Polynomials*, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1975.

Retrieve articles in *Mathematics of Computation*
with MSC:
65D30

Retrieve articles in all journals with MSC: 65D30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878702-4

Keywords:
Cauchy principal value integrals,
quadrature rules,
convergence

Article copyright:
© Copyright 1987
American Mathematical Society