Chebyshev-type integration rules of minimum norm

Authors:
Philip Rabinowitz and Nira Richter-Dyn

Journal:
Math. Comp. **24** (1970), 831-845

MSC:
Primary 65D30

DOI:
https://doi.org/10.1090/S0025-5718-1970-0298947-1

MathSciNet review:
0298947

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Equal-weight integration rules are studied in the context of certain families of Hilbert spaces of analytic functions defined in a family of confocal ellipses containing the interval of integration. Rules which minimize the norm of the error functional in these spaces are shown to exist and several such rules are tabulated. Asymptotic properties of these rules are studied for ellipses shrinking to the integration interval and for ellipses expanding to cover the entire plane. In the latter case, an algebraic formulation for these asymptotic rules is given and it is shown that they agree with the classical Chebyshev integration rules whenever such rules exist.

**[1]**R. E. Barnhill & J. A. Wixom, "Quadratures with remainders of minimum norm. II,"*Math. Comp.*, v. 21, 1967, pp. 382-387. MR**36**#6139. MR**0223090 (36:6139)****[2]**R. E. Barnhill, J. E. Dennis, Jr. & G. M. Nielson, "A new type of Chebyshev quadrature,"*Math. Comp.*, v. 23, 1969, pp. 437-441. MR**39**#3698. MR**0242367 (39:3698)****[3]**P. J. Davis,*Interpolation and Approximation*, Blaisdell, Waltham, Mass., 1963. MR**28**#393. MR**0157156 (28:393)****[4]**R. F. Fletcher & M. J. D. Powell, "A rapidly convergent descent method for minimization,"*Comput. J.*, v. 6, 1963/64, pp. 163-168. MR**27**#2096. MR**0152116 (27:2096)****[5]**M. Golomb & H. F. Weinberger,*Optimal Approximation and Error Bounds*, Proc. Sympos. Numerical Approximation (Madison, Wis., 1958) Univ. of Wisconsin Press, Madison, Wis., 1959, pp. 117-190. MR**22**#12697. MR**0121970 (22:12697)****[6]**F. B. Hildebrand,*Introduction to Numerical Analysis*, McGraw-Hill, New York, 1956. MR**17**, 788. MR**0075670 (17:788d)****[7]**P. Rabinowitz & N. Richter, "Asymptotic properties of minimal integration rules,"*Math. Comp.*, v. 24, 1970 pp. 593-609. MR**0298946 (45:7995)****[8]**N. Richter, "Properties of minimal integration rules,"*SIAM J. Numer. Anal.*, v. 6, 1969, pp. 67-79. MR**0260176 (41:4804)**

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/S0025-5718-1970-0298947-1

Keywords:
Chebyshev integration rules,
minimum norm rules,
norm of error functional,
Hilbert space,
analytic functions,
asymptotic integration rules,
equal-weight integration rules

Article copyright:
© Copyright 1970
American Mathematical Society