Equiconvergence and equisummability of nonharmonic Fourier expansions with ordinary trigonometric series

Abstract

Givenf e L(−π, π), we consider its nonharmonic Fourier series\(f(x) \sim \sum c_n e^{i\lambda _n x} \), where λn are the roots of the entire function L(z) = ∫ -π π e izt dσ (t). We show that this series is equiconvergent, uniformly inside (-π, π), and equisummable with the Fourier series off with respect to the trigonometric system if σ′ (t) =k (t) (π - ∣t∣)-α, α e (0, 1), vark <∞, k (π −0) ≠ 0,k (− π + 0) ≠ 0.