Almost orthogonality of Birkhoff's solutions
The paper concerns estimates for eigenfunctions for a quasi-differential expression \(I(y) \equiv D_ ny\) where \(D_ n\) is a superposition of first order linear differential expressions with coefficients from \(L(0,1)\). It is proved that a fundamental system of Birkhoff's solutions of an equation \(I(y)=\lambda y\) is almost orthogonal in \(L^ 2(0,1)\). Moreover, for an arbitrary solution \(y\) of the equation \(I(y)= \lambda y\) estimations of the \(L_ p\)-norm of \(D_ ky\) by the \(L_ 2\)-norm of \(D_ jy\) and suitable powers of \(| \rho |\) and \(| im \rho |\) where \(\rho=\lambda^{1 \over n}\) are obtained.
superposition of first order linear differential expressions
fundamental system of Birkhoff's solutions
scientific article; zbMATH DE number 484142