Optimal quadrature problem on Hardy–Sobolev classes

Abstract

Let H ˜ ∞ , β r denote those 2 π -periodic, real-valued functions f on R , which are analytic in the strip S β ≔ { z ∈ C : | Im z | < β } , β > 0 and satisfy the restriction | f ( r ) ( z ) | ⩽ 1 , z ∈ S β . Denote by [ x ] the integral part of x. We prove that the rectangular formula Q N * ( f ) = 2 π N ∑ j = 0 N - 1 f 2 π j N is optimal for the class of functions H ˜ ∞ , β r among all quadrature formulae of the form Q 2 N ( f ) = ∑ i = 1 n ∑ j = 0 ν i - 1 a ij f ( j ) ( t i ) , where the nodes 0 ⩽ t 1 < ⋯ < t n < 2 π and the coefficients a ij ∈ R are arbitrary, i = 1 , … , n , j = 0 , 1 , … , ν i - 1 , and ( ν 1 , … , ν n ) is a system of positive integers satisfying the condition ∑ i = 1 n 2 [ ( ν i + 1 ) / 2 ] ⩽ 2 N . In particular, the rectangular formula is optimal for the class of functions H ˜ ∞ , β r among all quadrature formulae of the form: Q N ( f ) = ∑ i = 1 N a i f ( t i ) , where 0 ⩽ t 1 < ⋯ < t N < 2 π and a i ∈ R , i = 1 , … , N . Moreover, we exactly determine the error estimate of the optimal quadrature formulae on the class H ˜ ∞ , β r .