On an inequality of Kolmogorov type for a second-order difference expression (Q1288859)

The authors study formally symmetric second order difference operators \(M\) on the weighted \(\ell ^2\) Hilbert space. They investigate the best constant \(K\) in the inequality \[ \|P_1Mx\|^2 \leq K\|x\|\|P_2M^2x\|, \] for all sequences \(x\) with only finitely many nonzero terms, where \(P_i\) cuts off the first \(i\) coordinates of the sequence. In the case when \(M\) is essentially selfadjoint the authors relate the constant \(K\) to the behaviour of the function \(m(z),\) the Cauchy transform of the orthogonality measure associated with the matrix \(M.\) Some examples and numerical computations are provided. It is worthwhile observing that considering the weighted \(\ell^2 \) space is redundant since by unitary equivalence the problem can be formulated in the classical \(\ell^2\) space setting. Moreover, by Schwarz's inequality the one in question holds with a constant \(K=1\) for sequences \(x\) whose first four terms vanish.