Error Estimate for a Quadrature Formula for H ² Functions

Abstract

Let f ∈ H ², where H ² is the Hardy space on the unit disc. Let −1 < x ₁ < x ₂ < ... < x _n < 1 be fixed given numbers. Consider ${\sup _{\left\| f \right\| \leqslant 1}}\left| {\int_{ - 1}^1 {f(t)dt - \sum\nolimits_{j = 1}^n {{a_j}f({x_j})} } } \right|: = \in ({a_1}, \ldots ,{a_n}): = \in (a)$. Here ‖ · ‖ is the norm in H ², a _j = const, a = (a ₁,...,a _n ). The quantity ∊ _n ≔ min _a ∊(a) is computed.