On the pointwise convergence of a class of nonharmonic Fourier series

Abstract:

Extending a classical theorem of Levinson [1, Theorem XVIII], we show that when the numbers $\left \{ {{\lambda _n}} \right \}$ are given by ${\lambda _n} = n + \tfrac {1}{4}(n > 0)$, ${\lambda _0} = 0$, and ${\lambda _{ - n}} = - {\lambda _n}(n > 0)$, each function $f$ in ${L^2}( - \pi ,\pi )$ has a unique nonharmonic Fourier expansion $f(t) \sim \sum \nolimits _{ - \infty }^\infty {{c_n}{e^{i{\lambda _n}t}}}$, which is equiconvergent with its ordinary Fourier series, uniformly on each closed subinterval of $( - \pi ,\pi )$.