Peculiarities of the numerical realization of unsaturated quadrature formulas on a finite interval
The author presents a sufficient condition for a weighted unsaturated quadrature formula to be well-conditioned. Let \(I=[-1,\,1]\). For a weight function \(r\in L_p(I)\) with \(1< p < \infty\) and \(f\in C(I)\), the author considers the quadrature formula \[ \int_{-1}^1 r(t)\,f(t)\,\text{d}t = \sum_{k=1}^n c_k\,f(t_k) + e_n(f)\,, \] where \(t_k\in I\) are the zeros of the Chebyshev polynomial \(T_n(t)\) or rather \((t^2 -1)\,U_{n-1}(t)\). The coefficients \(c_k\) are given by \[ c_k = \frac{1}{\omega_n^{\prime}(t_k)}\, \int_{-1}^1 r(t)\,\frac{\omega_n(t)}{t-t_k}\,\text{d}t \] with \(\omega_n(t) = (t-t_1)\,\ldots\,(t-t_n)\). For the norm of the error functional \(e_n\) one has \[ \|e_n\| = \int_{-1}^1 |r(t)|\,\text{d}t + \sum_{k=1}^n |c_k|\,. \] Then it is shown that this quadrature formula is unsaturated and well-conditioned, i.e., there exists a constant \(A\) not depending on \(n\) with \(\|e_n\| \leq A\). Further it holds \[ \lim_{n\to \infty} \sum_{k=1}^n |c_k| = \int_{-1}^1 |r(t)|\, \text{d}t\,. \] Numerical examples are not given.
scientific article; zbMATH DE number 6834936