On a moment problem and a class of perfect summability methods (Q1409821)

Theorem 1. Let \(D:= {d^n\over dt^n}+ \sum^{n-1}_{j=0} a_j(t) {d^j\over dt^j}\) be a differential operator with \(a_j\in L^1[0,1]\). Let \(f\in L^1[0,1]\) and \(\{\lambda_k\}\) be an increasing sequence of positive integers, \(\sum \lambda^{-1}_k= \infty\). For a certain \(\rho\in (0,1)\) assume \[ \int^1_0 D(t^{\lambda_k})f(t)\,dt= O(\lambda^{-n}_k p^{\lambda_k}),\quad k\to\infty. \] Then \(f(t)= 0\) a.e. on \(t\in (\rho, 1)\). Theorem 2 is the generalization of Włodarski's result. The ``continuous summability methods'' are perfect.