How slowly can quadrature formulas converge?

Abstract:

Let $\{ {Q_n}\} _{n = 1}^\infty$ denote a sequence of quadrature formulas, ${Q_n}(f) \equiv \sum _{i = 1}^{{k_n}}w_i^{(n)}f(x_i^{(n)})$, such that ${Q_n}(f) \to \int _0^1 f (x)dx$ for all $f \in C[0,1]$. Let $0 < \varepsilon < \frac {1}{4}$ and a sequence $\{ {a_n}\} _{n = 1}^\infty$ be given, where ${a_1} \geqq {a_2} \geqq {a_3} \geqq ...$, and where ${a_n} \to 0$ as $n \to \infty$. Then there exists a function $f \in C[0,1]$ and a sequence $\{ {n_k}\} _{k = 1}^\infty$ such that $|f(x)| \leqq 2{a_1}/|(1 - 4)|$, and such that $\int _0^1 f (x)dx - {Q_{{n_k}}}(f) = {a_{k,}}k = 1,2,3,...$.