TY - JOUR

T1 - On summation of nonharmonic Fourier series.

AU - Белов, Юрий Сергеевич

AU - Lyubarskii, Yurii

PY - 2016/12

Y1 - 2016/12

N2 - Let a sequence \Lambda \subset {\mathbb {C}} be such that the corresponding system of exponential functions {\mathcal {E}}(\Lambda ):=\left\{ {\text {e}}^{i\lambda t}\right\} _{\lambda \in \Lambda } is complete and minimal in L^2(-\pi ,\pi ), and thus each function f\in 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.

AB - Let a sequence \Lambda \subset {\mathbb {C}} be such that the corresponding system of exponential functions {\mathcal {E}}(\Lambda ):=\left\{ {\text {e}}^{i\lambda t}\right\} _{\lambda \in \Lambda } is complete and minimal in L^2(-\pi ,\pi ), and thus each function f\in 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.

M3 - Article

VL - 43

SP - 291

EP - 309

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

IS - 2

ER -