A Mercer-type theorem (Q1190056)

We present a generalization of Mercer's theorem. Theorem. Assume that positive sequences \((p_ n)\) and \((b_ n^{(q)})\) and nonnegative sequences \((b_ n^{(i)})\), \(i=1,2,\dots,q-1\), satisfy the condition \(P_ n=p_ 0+\dots+p_ n\to+\infty\) (\(n\to\infty\)); \(\varliminf_{n\to\infty}\bigl(b_ n^{(q)}-\sum_{i=1}^{q-1} b_ n^{(i)}\bigr)=b>0\), \(\sum_{i=1}^ q b_ n^{(i)}\leq H\) (\(n=0,1,\dots)\), where \(H\) is independent of \(n\), \[ {b_ n^{(q)} \over p_ n} \geq {b_ n^{(q-1)} \over p_{n-1}} \geq {b_ n^{(q-2)} \over p_{n_ 2}} \geq\dots\geq {b_ n^{(2)} \over p_{n-q+2}} \geq {b_ n^{(1)} \over p_{n-q+1}}. \] Then the transformation \[ t_ n={{1-\sum_{i=1}^ q b_ n^{(i)}} \over P_ n} \sum_{k=0}^ n p_ k S_ k +\sum_{i=1}^ q b_ n^{(i)} S_{n-q+i} \] is completely ineffective.