Some remarks on quadrature formulae (Q2895241)

Let \( Q_{n}(X,v,f)=\sum_{k=1}^n\lambda_{kn}(X,v)f(x_{kn})\) be an interpolatory quadrature (IQ) for \(\int_{-1}^{1} f(x)v(x)\,dx\). It is well known that NEWLINE\[NEWLINE\lim_{n\to\infty} Q_{n}(X,v,f)=\int_{-1}^1 f(x)v(x)\, dx,\quad(\forall)\,\,f\in C\tag{1}NEWLINE\]NEWLINE NEWLINE(\(C\) being the space of all continuous functions on \([-1,1]\)), if and only if \(\sum_{k=1}^n|\lambda_{kn}(X,v)|\leq A,\;n=1,2,\dots .\) The author considers the condition NEWLINE\[NEWLINE\lim_{n\to\infty}\sum_{\lambda_{kn}(v)<0}|\lambda_{kn}(v)|=0 NEWLINE\]NEWLINE NEWLINEand proves that (1) is true. These quadratures are called quasi positive interpolatory quadratures (qPIQ). This kind of quadratures are investigated. The author verifies certain generalizations of two conjectures raised, respectively, by G. Milovanovic and W. Gautschi.