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

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

MathSciNet review:
1361808

Full-text PDF

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. Badkov,*Convergence in the mean and almost everywhere of Fourier series in polynomials orthogonal on an interval*, Math. U.S.S.R. - Sb. 24 (1974), 223--256. MR**50:7938****[2]**A. Ghizzetti and A. Ossicini,*Quadrature Formulae*, Academic Press, New York (1970). MR**42:4012****[3]**G.H. Golub and J. Kautsky,*Calculation of Gauss quadrature with multiple free and fixed knots*, Numer. Math.**41**(1983), 147--162. MR**84i:65030****[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. Babe\c{s} - Bolyai Mathematica**37**(1992), 47--63. MR**95j:65020****[5]**L. Gori and E. Santi,*On the convergence of Turán type rules for Cauchy principal value integrals*, Calcolo**28**(1991), 21--35. MR**94e:65026****[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,*Remarks on mechanical quadrature, (Russian)*, Acta Math. Acad. Sci. Hungaricae**8**(1957), 473--476. MR**20:196****[8]**C.A. Micchelli,*The fundamental theorem of algebra for monosplines with multiplicities*, in Linear Operators and Approximation, eds. P. Butzer and J.P. Kahane, B. Sz. Nagy, ISNM 20, Birkhäuser Verlag, 1971, pp. 372--379. MR**52:14758****[9]**C.A. Micchelli and T.J. Rivlin,*Turán formulas and highest precision quadrature rules for Chebyshev coefficients*, IBM Journal of Research and Development**16**(1972), 372--379. MR**48:12784****[10]**C.A. Micchelli and A. Sharma,*On a problem of Turán : multiple node Gaussian quadrature*, Rend. Mat. VII**3**(1983), 529--552. MR**86d:41032****[11]**G.V. Milovanovi\'{c},*Construction of s-orthogonal polynomials and Turán quadratures*, in Approx. Theory III, Ni\v{s}, 1987, (ed. G.V. Milovanovi\'{c}), Univ. Ni\v{s}, 1988, pp. 311--328. MR**89g:65023****[12]**P. Nevai,*Mean convergence of Lagrange interpolation III*, Trans. A.M.S.**282**(1984), 669--698. MR**85c:41009****[13]**P. Turán,*On the theory of mechanical quadrature*, Acta Sci. Math. Szeged**2**(1950), 30--37. MR**12:164b****[14]**------,*On some open problems in approximation theory*, J. Approx. Theory**29**(1980), 23--85. MR**82e:41003**

Retrieve articles in *Mathematics of Computation*
with MSC (1991):
65D32,
41A55

Retrieve articles in all journals with MSC (1991): 65D32, 41A55

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:
https://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