|

Sufficient Conditions for Synthesability of Exponential Systems

Authors: Belov Yu.S. Published: 12.04.2018
Published in issue: #2(77)/2018  
DOI: 10.18698/1812-3368-2018-2-4-11

 
Category: Mathematics | Chapter: Substantial Analysis, Complex and Functional Analysis  
Keywords: exponential systems, spectral synthesis, hereditary completenness

The study considers the sufficient conditions for synthesability of exponential systems {eiλt}λ∈Λ in L2(-π, π) in terms of generating function G = GΛ. The obtained results are of interest not only for the theory of functions but also for the theory of operators, since the synthesizability of ε(Λ) systems corresponds to the synthesizability of special perturbations of the level one of self-adjoint operators

References

[1] Young R.M. On complete biorthogonal system. Proc. Amer. Math. Soc., 1981, vol. 83, no. 3, pp. 537–540. DOI: 10.2307/2044113 Available at: https://www.jstor.org/stable/2044113

[2] Baranov A., Belov Y., Borichev A., Yakubovich D. Recent developments in spectral synthesis for exponential systems and for nonselfadjoint operators. Recent Trends in Analysis. Proc. of the Conf. in Honor of Nikolai Nikolski. Bucharest, Theta Foundation, 2013. Pp. 17–34.

[3] Baranov A.D., Yakubovich D.V. Completeness and spectral synthesis of nonselfadjoint one-dimensional perturbations of selfadjoint operators. Adv. Math., 2016, vol. 302, pp. 740–798. DOI: 10.1016/j.aim.2016.07.020

[4] Markus A.S. The problem of spectral synthesis for operators with point spectrum. Mathematics of the USSR-Izvestiya, 1970, vol. 4, no. 3, pp. 670–696. DOI: 10.1070/IM1970v004n03ABEH000926

[5] Nikolskii N.K. Complete extensions of volterra operators. Mathematics of the USSR-Izvestiya, 1969, vol. 3, no. 6, pp. 1271–1276. DOI: 10.1070/IM1969v003n06ABEH000846

[6] Baranov A., Belov Y., Borichev A. Hereditary completeness for systems of exponentials and reproducing kernels. Adv. Math., 2013, vol. 235, pp. 525–554. DOI: 10.1016/j.aim.2012.12.008

[7] Levin B.Ya. Lectures on entire functions, translations of mathematical monographs. AMS, 1996. 248 p.

[8] Khrushchev S., Nikolski N., Pavlov B. Unconditional bases of exponentials and reprodu-cing kernels. In: Complex Analysis and Spectral Theory. Springer, 1981. Pp. 214–235.

[9] Gubreev G.M., Tarasenko A.A. Spectral decomposition of model operators in de Branges spaces. Sbornik: Mathematics, 2010, vol. 201, no. 11, pp. 1599–1634. DOI: 10.1070/SM2010v201n11ABEH004124

[10] Belov Y., Lyubarskii Y. On summation of nonharmonic Fourier series. Constructive approximation, 2016, vol. 43, iss. 2, pp. 291–309. DOI: 10.1007/s00365-015-9290-6

[11] Baranov A., Belov Y., Borichev A. Spectral synthesis in de Branges spaces. Geom. Funct. Anal., 2015, vol. 25, iss. 2, pp. 417–452. DOI: 10.1007/s00039-015-0322-y

[12] Lyubarskii Yu., Seip K. Complete interpolating sequences for Paley — Wiener spaces and Muckenhoupts condition. Rev. Mat. Iberoamericana, 1997, vol. 13, no. 2, pp. 361–376.

[13] Belov Yu.S. Model functions with nearly prescribed modulus. St. Petersburg Math. J., 2009, vol. 20, no. 2, pp. 163–174. DOI: 10.1090/S1061-0022-09-01042-5