Nonharmonic Fourier series and spectral theory

Abstract:

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