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Transactions of the American Mathematical Society

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Nonharmonic Fourier series and spectral theory

Author: Harold E. Benzinger
Journal: Trans. Amer. Math. Soc. 299 (1987), 245-259
MSC: Primary 42A65; Secondary 34B25, 42A20, 47B38
MathSciNet review: 869410
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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\vert {{\lambda _n} - n} \right\vert \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.

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