On weight functions which admit explicit Gauss-Turán quadrature formulas

Authors:
Laura Gori and Charles A. Micchelli

Journal:
Math. Comp. **65** (1996), 1567-1581

MSC (1991):
Primary 65D32; Secondary 41A55

MathSciNet review:
1361808

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Abstract | References | Similar Articles | Additional Information

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 -orthogonal polynomials, of the same degree, are independent of . These weights too are introduced and discussed here. Moreover, highest-precision quadratures for evaluating Fourier-Chebyshev coefficients are given.

**[1]**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****[2]**A. Ghizzetti and A. Ossicini,*Quadrature formulae*, Academic Press, New York, 1970. MR**0269116****[3]**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**, 10.1007/BF01390210**[4]**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****[5]**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**, 10.1007/BF02575867**[6]**------,*On the evaluation of Hilbert transforms by means of a particular class of Turán quadrature rules*, Numer. Algor.**10**(1995), 17--39.**[7]**O. Kis,*Remark on mechanical quadrature*, Acta Math. Acad. Sci. Hungar**8**(1957), 473–476 (Russian). MR**0093673****[8]**Charles Micchelli,*The fundamental theorem of algebra for monosplines with multiplicities*, Linear operators and approximation (Proc. Conf., Oberwolfach, 1971), Birkhäuser, Basel, 1972, pp. 419–430. Internat. Ser. Numer. Math., Vol. 20. MR**0393951****[9]**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**0334465****[10]**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****[11]**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****[12]**Paul Nevai,*Mean convergence of Lagrange interpolation. III*, Trans. Amer. Math. Soc.**282**(1984), no. 2, 669–698. MR**732113**, 10.1090/S0002-9947-1984-0732113-4**[13]**P. Turán,*On the theory of the mechanical quadrature*, Acta Sci. Math. Szeged**12**(1950), no. Leopoldo Fejer et Frederico Riesz LXX annos natis dedicatus, Pars A, 30–37. MR**0036797****[14]**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**, 10.1016/0021-9045(80)90138-0

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Additional Information

**Laura Gori**

Affiliation:
Dipartimento di Metodi e Modelli Matematici, per le Scienze Applicate, Università “La Sapienza", Via Antonio Scarpa , 16-00161 Roma, Italia

**Charles A. Micchelli**

Affiliation:
Mathematical Sciences Department, IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598

DOI:
http://dx.doi.org/10.1090/S0025-5718-96-00769-7

Keywords:
Quadrature,
Tur\'{a}n-type integration rules,
generalized Jacobi weights

Received by editor(s):
November 29, 1994

Received by editor(s) in revised form:
August 9, 1995

Additional Notes:
The second author was partially supported by the Alexander von Humboldt Foundation.

The first author was supported by Ministero Università e Ricerca Scientifica e Tecnologica – Italia.

Article copyright:
© Copyright 1996
American Mathematical Society