Localization principle for expansions of generalized functions with respect to the eigenfunctions of the Sturm-Liouville operator on a finite interval (Q1177775)

Let \(A\) be a differential operator generated from the expression \(-d^ 2/dx^ 2+q(x)\) on the interval \((0,\pi)\), where \(q(x)\) is an infinitely differentiable real function in \([0,\pi]\). The domain of definition \(D(A)\) of the operator \(A\) is the set of functions \(y(x)\in C^ 2[0,\pi]\) which satisfy the boundary conditions \(y(0)\cos \alpha+y'(0)\sin \alpha=0\), \(y(\pi)\cos \beta+y'(\pi)\sin \beta =0\). \(D_ p\) is the domain of definition of the operator \(A^ p\), \(p=0,1,\dots\) with the norm \(| y|_ p=\max_{x\in[0,\pi],j=0,1,\dots,p}|(A^ jy)(x)|\). \(D_ \infty\) is the set \(\bigcap^ \infty_{p=1}D_ p\) with projective limit topology. The space of generalized functions \(D_ \infty'\) is the set of all linear continuous functions on \(D_ \infty\). Let \(\{\lambda_ n(\xi)\}=\Lambda\), \(n=0,1,\dots\) be a set of real numbers. The quantities \[ S_ \xi(F,x)=\sum^ \infty_{n=0}\lambda_ n(\xi)C_ n(F)u_ n(x) \] are called the means of summation method of \(\Lambda\), where \(C_ n(F)=\langle F,u_ n\rangle\) is the operation of the functional \(F\in D_ \infty'\) on the function \(u_ n\). \(\Lambda\) is called the summation method of \({\mathcal L}(D_ \infty)\) type if for any generalized function \(F\), which is equal to zero on \((a,b)\), and for any \(\varepsilon > 0\) the means uniformly converge to zero for \(\xi\to \xi_ 0\), \(x\in [a+\varepsilon,b-\varepsilon]\). The author proves the following theorem: If for all \(p=0,1,2,\dots\) and \(\sigma>0\): \[ \max_{x\in[\sigma;2\pi-\sigma]}| (1/2+\sum^ \infty_{n=1}\lambda_ n(\xi)\cos nx)^{(p)}|\to 0,\quad\xi\to \xi_ 0, \] then \(\Lambda\) is the summation method of \({\mathcal L}(D_ \infty)\) type.