On Summation of Nonharmonic Fourier Series Article First Online: 19 May 2015 Received: 21 May 2014 Accepted: 21 January 2015 DOI :
10.1007/s00365-015-9290-6

Cite this article as: Belov, Y. & Lyubarskii, Y. Constr Approx (2016) 43: 291. doi:10.1007/s00365-015-9290-6 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.

Keywords Nonharmonic Fourier series Summation methods Paley–Wiener spaces Muckenhoupt condition Lagrange interpolation Communicated by Vilmos Totik.

The first author was supported by the Chebyshev Laboratory (St. Petersburg State University) under RF Government Grant 11.G34.31.0026, by JSC “Gazprom Neft,” and by RFBR Grants 12-01-31492 and 14-01-31163. The second author is partly supported by the Norwegian Research Council project DIMMA #213638. Part of this work was done while the authors were staying at the Center for Advanced Study, Norwegian Academy of Science, and they would like to express their gratitude to the institute for the hospitality.

References 1.

Baranov, A., Belov, Y., Borichev, A.: Hereditary completeness for systems of exponentials and reproducing kernels. Adv. Math.

235 , 525–554 (2013)

MathSciNet CrossRef MATH Google Scholar 2.

Gubreev, G., Tarasenko, A.: Spectral decomposition of model operators in de Branges spaces. Math. Sb. (Russian)

201 (11), 41–76 (2010). translation in Sb. Math. 201 (2010), no. 11–12, 1599–1634

MathSciNet CrossRef MATH Google Scholar 3.

Hruscev, S.V., Nikolskii, N.K., Pavlov, B.S.: Unconditional bases of exponentials and of reproducing kernels. In: Havin, V.P., Nikolski, N.K. (eds.) Complex Analysis and Spectral Theory (Leningrad, 1979/1980). Lecture Notes in Mathematics, vol. 864, pp. 214–335. Springer, Berlin, New York (1981)

4.

Levin, B.Y.: Lectures on Entire Functions, Translations of Mathematical Monographs, vol. 150. AMS, Providence (1996)

Google Scholar 5.

Lyubarskii, Yu., Seip, K.: Complete interpolating sequences for Paley–Wiener spaces and Muckenhoupt’s

\((A_p)\) condition. Rev. Math. Iberoam.

13 (2), 361–376 (1997)

MathSciNet CrossRef MATH Google Scholar 6.

Minkin, A.M.: Reflection of exponents and unconditional bases of exponentials. St. Petersburg Math. J.

3 , 1043–1068 (1992)

MathSciNet MATH Google Scholar 7.

Pavlov, B.S.: The basis property of a system of exponentials and the condition of Muckenhoupt. (Russian). Dokl. Akad. Nauk SSSR 247 , 37–40 (1979). English translation in Soviet Mathematics Doklady 20 (1979)

8.

Vasjunin, V.: Bases from eigensubspaces, and nonclassical interpolation problems. Funkc. Anal. Prilozhen (Russian)

9 (4), 65–66 (1975)

MathSciNet Google Scholar 9.

Young, R.: On complete biorthogonal system. Proc. Am. Math. Soc.

83 (3), 537–540 (1981)

CrossRef MATH Google Scholar © Springer Science+Business Media New York 2015

Authors and Affiliations 1. Chebyshev Laboratory St. Petersburg State University St. Petersburg Russia 2. Department of Mathematical Sciences Norwegian University of Science and Technology Trondheim Norway