On weight functions which admit explicit Gauss-Turán quadrature formulas

Gori, Laura; Micchelli, Charles

doi:10.1090/S0025-5718-96-00769-7

On weight functions which admit explicit Gauss-Turán quadrature formulas
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by Laura Gori and Charles A. Micchelli PDF

Math. Comp. 65 (1996), 1567-1581 Request permission

Abstract:

The main purpose of this paper is the construction of explicit Gauss-Turán quadrature formulas: they are relative to some classes of weight functions, which have the peculiarity that the corresponding $s$-orthogonal polynomials, of the same degree, are independent of $s$. These weights too are introduced and discussed here. Moreover, highest-precision quadratures for evaluating Fourier-Chebyshev coefficients are given.

References

V. M. Badkov, Convergence in the mean and almost everywhere of Fourier series in polynomials that are orthogonal on an interval, Mat. Sb. (N.S.) 95(137) (1974), 229–262, 327 (Russian). MR 0355464
A. Ghizzetti and A. Ossicini, Quadrature formulae, Academic Press, New York, 1970. MR 0269116
G. H. Golub and J. Kautský, Calculation of Gauss quadratures with multiple free and fixed knots, Numer. Math. 41 (1983), no. 2, 147–163. MR 703119, DOI 10.1007/BF01390210
L. Gori and M. L. Lo Cascio, A note on a class of Turán type quadrature formulas with generalized Gegenbauer weight functions, Studia Univ. Babeş-Bolyai Math. 37 (1992), no. 1, 47–63 (English, with English and Romanian summaries). MR 1324767
L. Gori Nicolò-Amati and E. Santi, On the convergence of Turán type rules for Cauchy principal value integrals, Calcolo 28 (1991), no. 1-2, 21–35 (1992). MR 1177934, DOI 10.1007/BF02575867
—, On the evaluation of Hilbert transforms by means of a particular class of Turán quadrature rules, Numer. Algor. 10 (1995), 17–39.
O. Kis, Remark on mechanical quadrature, Acta Math. Acad. Sci. Hungar. 8 (1957), 473–476 (Russian). MR 93673, DOI 10.1007/BF02020330
Charles Micchelli, The fundamental theorem of algebra for monosplines with multiplicities, Linear operators and approximation (Proc. Conf., Math. Res. Inst., Oberwolfach, 1971) Internat. Ser. Numer. Math., Vol. 20, Birkhäuser, Basel, 1972, pp. 419–430. MR 0393951
C. A. Micchelli and T. J. Rivlin, Turán formulae and highest precision quadrature rules for Chebyshev coefficients, IBM J. Res. Develop. 16 (1972), 372–379. Mathematics of numerical computation. MR 334465, DOI 10.1147/rd.164.0372
Charles A. Micchelli and A. Sharma, On a problem of Turán: multiple node Gaussian quadrature, Rend. Mat. (7) 3 (1983), no. 3, 529–552 (English, with Italian summary). MR 743396
Gradimir V. Milovanović, Construction of $s$-orthogonal polynomials and Turán quadrature formulae, Numerical methods and approximation theory, III (Niš, 1987) Univ. Niš, Niš, 1988, pp. 311–328. MR 960351
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
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
P. Turán, On some open problems of approximation theory, J. Approx. Theory 29 (1980), no. 1, 23–85. P. Turán memorial volume; Translated from the Hungarian by P. Szüsz. MR 595512, DOI 10.1016/0021-9045(80)90138-0