Characterization of algebraic curves by Chebyshev quadrature

Complex potential theory is used to show that Chebyshev-type quadrature works particularly well on algebraic Jordan curves Γ in ℝ ^d , supplied with normalized arc length or a similar probability measure μ. Evaluating the integral ∫_Γ fdμ by the arithmetic mean of the value off on any cycle ofN equally spaced nodes on Γ (relative to μ), the quadrature error will, be bounded byAe ^−bN sup_Γ|f| for allN and all polynomialsf(x) of degree ≤cN. It is plausible that small shifts of the nodes would give quadrature error zero for such polynomials. There are related results for algebraic Jordan arcs and certain algebraic surfaces. The situation is completely different for nonalgebraic curves and surfaces, where corresponding quadrature remainders are at least of order 1/N.