On Summation of Nonharmonic Fourier Series |
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Authors: | Yurii Belov Yurii Lyubarskii |
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Institution: | 1.Chebyshev Laboratory,St. Petersburg State University,St. Petersburg,Russia;2.Department of Mathematical Sciences,Norwegian University of Science and Technology,Trondheim,Norway |
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Abstract: | 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. |
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