On the equiconvergence of expansions by Riesz bases formed by eigenfunctions of a linear differential operator of order 2n (Q1057414)

By a classical result of A. Haar, for every square-integrable function, the difference of its trigonometrical Fourier series and of its Fourier series corresponding to the orthonormal Legendre system converges to zero uniformly on every compact subset of the open basic interval. By a recent result of I. Joó and the author the same assertion holds if we replace the Legendre system by an arbitrary complete orthonormal system (or more generally, by an arbitrary Riesz basis) consisting of eigenfunctions of an operator \(-u''+qu\) where q is a locally integrable complex-valued function. The present paper generalizes this result to the operator \((- 1)^ nu^{(2n)}+qu\) where \(n=1,2,..\). and q is a locally square- integrable complex-valued function. A more general result (concerning the operator \(u^{(n)}+q_ 2u^{(n-2)}+...+q_ nu\), \(n=1,2,...)\) is stated in C. R. Acad. Sci. Paris, Ser. I 299, 217-219 (1984); its proof will also appear in the Acta Math. Hung.