Abstract

In the Sobolev space L^(m)_2 (0, 1) optimal quadrature formulas of the form \int_0^1ϕ(x)dx \cong \sum_{\beta=0}^NC_βϕ(x_β) with the nodes x_i = ηih, xN−i = 1 − ηih, i = 0, t − 1, 0 ≤ η0 < η1 <· · · < ηt−1 < t, t ∈ N, xβ = hβ, 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.