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Given a class S
of functions that are Riemann integrable on [0,1]. A quadrature formula
∫
01f(x)dx = ∑
i=1∞aif(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)1∕2 f(x)dx provided we allow the abscissas xi to take
on complex values.
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