On Chebyshev-Type Quadratures

According to a result of S. N. Bernstein, n-point Chebyshev quadrature for- mulas, with all nodes real, do not exist when n = 8 or n _ 10. Modifications of such quad- rature formulas have recently been suggested by R. E. Barnhill, J. E. Dennis, Jr. and G. M. Nielson, and by D. Kahaner. We establish here certain empirical observations made by these authors, notably the presence of multiple nodes. We also show how some of the quadrature rules proposed can be constructed by solving single algebraic equations, and we compute the respective nodes to 25 decimal digits. The same formulas also arise in recent work of P. Rabinowitz and N. Richter as limiting cases of optimal Chebyshev-type quad- rature rules in a Hilbert space setting.