On a Conjecture of D. B. Hunter
نویسنده
چکیده
We consider errors of positive quadrature formulas applied to Chebyshev polynomials. These errors play an important role in the error analysis for many function classes. Hunter conjectured that the supremum of all errors in Gaussian quadrature of Chebyshev polynomials equals the norm of the quadrature formula. We give examples, for which Hunter's conjecture does not hold. However, we prove that the conjecture is valid for all positive quadrature if the supremum is replaced by the limes superior. Considering a xed positive quadrature formula and the sequence of all Chebyshev polynomials, we show that large errors are rare. Let R n be the remainder of a positive quadrature formula on ?1; 1]. This means that R n is a linear functional on C?1; 1] of the form R n f] := Z 1 ?1 f(x)w(x) dx ? n X =1 a f(x); where w 2 L 1 ?1; 1] , w 0 , a > 0 and ?1 x 1 < x 2 < : : : < x n 1. In the particularly interesting case that the underlying quadrature formula is a Gaussian formula with respect to w , we write R G n instead of R n .
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