Nonharmonic Fourier series and spectral theory

Abstract

We consider the problem of using functions g n ( x ) := e x p ( i λ n x ) {g_n}(x): = exp(i{\lambda _n}x) to form biorthogonal expansions in the spaces L p ( − π , π ) {L^p}( - \pi ,\,\pi ) , for various values of p p . The work of Paley and Wiener and of Levinson considered conditions of the form | λ n − n | ≤ Δ ( p ) \left | {{\lambda _n} - n} \right | \leq \Delta (p) which insure that { g n } \{ {g_n}\} is part of a biorthogonal system and the resulting biorthogonal expansions are pointwise equiconvergent with ordinary Fourier series. Norm convergence is obtained for p = 2 p = 2 . In this paper, rather than imposing an explicit growth condition, we assume that { λ n − n } \{ {\lambda _n} - n\} is a multiplier sequence on L p ( − π , π ) {L^p}( - \pi ,\,\pi ) . Conditions are given insuring that { g n } \{ {g_n}\} inherits both norm and pointwise convergence properties of ordinary Fourier series. Further, λ n {\lambda _n} and g n {g_n} are shown to be the eigenvalues and eigenfunctions of an unbounded operator Λ \Lambda which is closely related to a differential operator, i Λ i\Lambda generates a strongly continuous group and − Λ 2 - {\Lambda ^2} generates a strongly continuous semigroup. Half-range expansions, involving cos λ n x {\text {cos}}{\lambda _n}x or sin λ n x {\text {sin}}{\lambda _n}x on ( 0 , π ) (0,\,\pi ) are also shown to arise from linear operators which generate semigroups. Many of these results are obtained using the functional calculus for well-bounded operators.