On Summation of Nonharmonic Fourier Series

Let a sequence (Lambda subset {mathbb {C}}) be such that the corresponding system of exponential functions ({mathcal {E}}(Lambda ):=left{ {text {e}}^{ilambda t}right} _{lambda in Lambda }) is complete and minimal in (L^2(-pi ,pi )), and thus each function (fin L^2(-pi ,pi )) corresponds to a nonharmonic Fourier series in ({mathcal {E}}(Lambda )). We prove that if the generating function (G) of (Lambda ) satisfies the Muckenhoupt ((A_2)) condition on ({mathbb {R}}), then this series admits a linear summation method. Recent results show that the ((A_2)) condition cannot be omitted.