Hilbert and Bessel properties of systems of root functions of \(2m\)th-order operators (Q5951176)

A system \(\{e_n\}_{n=1}^\infty\) of elements in the Hilbert space \({\mathbb H}\) is called Hilbert (respectively Bessel) system if \[ \exists\;\alpha>0\;\;\forall f\in{\mathbb H}\;\;\sum_{n=1}^\infty | (f,e_n)| ^2\geq\alpha\| f\| ^2 \] \[ \left( \text{respectively }\exists\;\beta>0\;\;\forall f\in{\mathbb H}\;\;\sum_{n=1}^\infty | (f,e_n)| ^2\leq\beta\| f\| ^2\right). \] These conditions are important in various problems of (numerical) approximation related, for example, to systems of eigenfunctions of a differential operator. Here a little bit more general objects, the so-called root functions, are used. A natural space in this case is \({\mathbb H}=L^2(a,b)\). \vskip .1 in Two kinds of ordinary differential operators are considered here: \[ L_1\tilde u=\tilde u^{(2m)} \] and \[ L_2u=u^{(2m)}+p_2(x)u^{(2m-2)}+ p_3(x)u^{(2m-3)}+\cdots+p_{2m-1}(x)u'+p_{2m}(x)u. \] The main result of the present paper consists in stating and proving (very technical) sufficient conditions implying that if one of the two systems \[ \{u_{n,p}/\| u_{n,p}\| \}\quad \text{and}\quad \{\widetilde {u}_{n,p}/\| \widetilde {u}_{n,p}\| \} \] of normalized root functions is Hilbert or Bessel, then the same holds for the other one. Here it should be explained why double indexation is used: the root functions are very similar to the eigenfunctions, hence the first index corresponds to the number of the eigenvalue, while the second one is related to its multiplicity. Relations between corresponding coefficients \(\alpha\), \(\widetilde\alpha\) and \(\beta\), \(\widetilde\beta\) are also given.