A note on a paper by G. Mastroianni and G. Monegato

Criscuolo, G.

doi:10.1090/S0025-5718-03-01540-0

A note on a paper by G. Mastroianni and G. Monegato
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Math. Comp. 73 (2004), 243-250 Request permission

Abstract:

Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate the integral \[ \int _a^b \int _{-1}^1\frac {f(x,y)}{x-y}dxdy, \] where $(a,b)\equiv (-1,1)$ or $(a,b)\equiv (a,-1)$ or $(a,b)\equiv (1,b)$ and $f(x,y)$ is a smooth function (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 2001, 251–267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper.

References

Giuliana Criscuolo, A new algorithm for Cauchy principal value and Hadamard finite-part integrals, J. Comput. Appl. Math. 78 (1997), no. 2, 255–275. MR 1433509, DOI 10.1016/S0377-0427(96)00142-2
Giuliana Criscuolo and Giuseppe Mastroianni, Mean and uniform convergence of quadrature rules for evaluating the finite Hilbert transform, Progress in approximation theory, Academic Press, Boston, MA, 1991, pp. 141–175. MR 1114771
Giuliana Criscuolo, Giuseppe Mastroianni, and Paul Nevai, Associated generalized Jacobi functions and polynomials, J. Math. Anal. Appl. 158 (1991), no. 1, 15–34. MR 1113396, DOI 10.1016/0022-247X(91)90263-Y
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
G. Mastroianni and M. G. Russo, Lagrange interpolation in some weighted uniform spaces, Facta Univ. Ser. Math. Inform. 12 (1997), 185–201. Dedicated to Professor Dragoslav S. Mitrinović (1908–1995) (Niš, 1996). MR 1644840
G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 (2001), no. 233, 251–267. MR 1803127, DOI 10.1090/S0025-5718-00-01272-2
Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, DOI 10.1090/memo/0213
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517