Vol. 15, No. 3, 1965

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Simple quadratures in the complex plane

Philip J. Davis

Vol. 15 (1965), No. 3, 813–824

Given a class S of functions that are Riemann integrable on [0,1]. A quadrature formula 01f(x)dx = i=1aif(xi) is called a simple quadrature for S if the xi are distinct and if both the ai and the xi are fixed and independent of the particular function of S selected. It is known that if S is too large, for example if S = C[0,1], a simple quadrature cannot exist. On the other hand, if S is sufficiently restricted, for example the class of all polynomials, then simple quadratures exist.

The present paper investigates further the existence of simple quadratures. It is proved among other things that if S is the class of analytic functions that are regular in the closure of an ellipse with foci at ±1, a simple quadrature exists for the weighted integral 1+1(1 x2)12 f(x)dx provided we allow the abscissas xi to take on complex values.

Mathematical Subject Classification
Primary: 41.44
Secondary: 30.35
Received: 25 April 1964
Published: 1 September 1965
Philip J. Davis