Abstract
Let the integers ( v k ) 1 n , 1 ⩽ v k ⩽ r, be fixed. We show that there exists a quadrature formula with nodes a < x 1 ∗ < … < x n ∗ < b of multiplicities v 1, …, v n , respectively, which has a minimal error in the Sobolev space W∞ r[a, b] among all quadratures with nodes ( x k ) 1 n , a ⩽ x 1 < … < x n ⩽ b, of the same multiplicities ( v k ) 1 n .
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