Optimal quadrature problem on n-information for Hardy-Sobolev classes

For β > 0 and an integer r ≥ 2, denote by [(H)tilde]_￥,b^rtilde H_{infty ,beta }^r those 2π-periodic, real-valued functions f on ℝ, which are analytic in S _β := {z ∈ ℂ: |Im z| < β} and satisfy the restriction |f ^(r)(z)|≤1, z ∈ S _β . The optimal quadrature formulae about information composed of the values of a function and its kth (k = 1, ..., r − 1) derivatives on free knots for the classes [(H)tilde]_￥,b^rtilde H_{infty ,beta }^r are obtained, and the error estimates of the optimal quadrature formulae are exactly determined.