Optimal quadrature formulas with positive coefficients in space

In the Sobolev space L 2 ( m ) ( 0 , 1 ) optimal quadrature formulas of the form ∫ 0 1 φ ( x ) d x ≅ ∑ β = 0 N C β φ ( x β ) with the nodes x i = η i h , x N − i = 1 − η i h , i = 0 , t − 1 ¯ , 0 ≤ η 0 < η 1 < ⋯ < η t − 1 < t , t ∈ N , x β = h β , t ≤ β ≤ N − t , h = 1 N are investigated. For optimal coefficients C β explicit forms are obtained and the norm of the error functional is calculated for any natural numbers m and N . In particular, in the case t = 1 and η 0 = 0.205 for m = 2 , 3 , … , 14 optimal quadrature formulas with positive coefficients are numerically obtained and some of them are compared with well-known optimal formulas.