We consider the problem of using functions gn(x):= exp(O'nx) to form biorthogonal expansions in the spaces LP(-'lT, 'IT), for various values of p. The work of Paley and Wiener and of Levinson considered conditions of the form IAn -nl .:; t. (p) which insure that {g"} 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

more »

... dition, we assume that {A" -n} is a multiplier sequence on LP(-'1T, 'IT). Conditions are given insuring that {gn} inherits both norm and pointwise convergence properties of ordinary Fourier series. Further, An and g" are shown to be the eigenvalues and eigenfunctions of an unbounded operator A which is closely related to a differential operator, i A generates a strongly continuous group and _A2 generates a strongly continuous semigroup. Half-range expansions, involving cos A" x or sin An X on (0, 'IT) 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.