Systems biorthogonal to exponential systems on a finite union of intervals. (English) Zbl 1523.46006
Summary: We study the properties of a system biorthogonal to a complete and minimal system of exponentials in \(L^2(E)\), where \(E\) is a finite union of intervals, and show that in the case when \(E\) is a union of two or three intervals the biorthogonal system is also complete.
MSC:
| 46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
exponential system; Paley-Wiener space; disconnected spectrum; biorthogonal system; Young theoremReferences:
| [1] | Avdonin, S.A.: On the question of Riesz bases of exponential functions in \(L^2\). Vestnik Leningrad. Univ. 13, 5-12 (1974). (in Russian) · Zbl 0296.46033 |
| [2] | Baranov, A.; Belov, Yu, Systems of reproducing kernels and their biorthogonal: completeness or incompleteness?, Int. Math. Res. Notices, 22, 5076-5108 (2011) · Zbl 1239.46012 |
| [3] | Baranov, A.; Belov, Y.; Borichev, A., The Young type theorem in weighted Fock spaces, Bull. Lond. Math. Soc., 50, 2, 357-363 (2018) · Zbl 1395.30004 · doi:10.1112/blms.12144 |
| [4] | Belov, Y., Uniqueness of Gabor series, Appl. Comput. Harm. Anal., 39, 3, 545-551 (2015) · Zbl 1325.30026 · doi:10.1016/j.acha.2015.03.006 |
| [5] | Belov, Y.; Borichev, A.; Kuznetsov, A., Upper and lower densities of Gabor Gaussian systems, Appl. Comput. Harm. Anal., 49, 2, 438-450 (2020) · Zbl 1446.42042 · doi:10.1016/j.acha.2020.05.003 |
| [6] | Beurling, A.; Malliavin, P., On the closure of characters and the zeros of entire functions, Acta Math., 118, 79-93 (1967) · Zbl 0171.11901 · doi:10.1007/BF02392477 |
| [7] | Boas, RP, Entire Functions (1954), New York: Academic Press, New York · Zbl 0058.30201 |
| [8] | Fricain, E., Complétude des noyaux reproduisants dans les espaces modèles, Ann. Inst. Fourier (Grenoble), 52, 2, 661-686 (2002) · Zbl 1032.46040 · doi:10.5802/aif.1897 |
| [9] | Kozma, G.; Nitzan, S., Combining Riesz bases, Invent. Math., 199, 1, 267-285 (2015) · Zbl 1309.42048 · doi:10.1007/s00222-014-0522-3 |
| [10] | Kozma, G., Nitzan, S., Olevskii, A.: A set with no Riesz basis of exponentials, arXiv:2110.02090 |
| [11] | Khruschev, S.V., Nikolski, N.K., Pavlov, B.S.: Unconditional bases of exponentials and reproducing kernels. In: Complex Analysis and Spectral Theory. Lecture Notes in Mathematics, vol. 864. Springer, Berlin, pp. 214-335 (1981) · Zbl 0466.46018 |
| [12] | Landau, HJ, A sparse regular sequence of exponentials closed on large sets, Bull. Am. Math. Soc., 70, 566-569 (1964) · Zbl 0131.06401 · doi:10.1090/S0002-9904-1964-11202-7 |
| [13] | Levin, B.Y.: Lectures on Entire Functions. Transl. Math. Monogr. vol. 150. AMS, Providence, RI (1996) · Zbl 0856.30001 |
| [14] | Minkin, A.M.: The reflection of indices and unconditional bases of exponentials. Algebra i Analiz3(5), 109-134 (1991); English transl.: St. Petersburg Math. J. 3(5), 1043-1068 (1992) · Zbl 0791.42021 |
| [15] | Nitzan, S., Olevskii, A.: Revisiting Landau’s density theorems for Paley-Wiener spaces. C. R. Acad. Sci. Paris, Ser. I 350, 509-512 (2012) · Zbl 1248.41010 |
| [16] | Olevskii, A.; Ulanovskii, A., Interpolation in Bernstein and Paley-Wiener spaces, J. Funct. Anal., 256, 10, 3257-3278 (2009) · Zbl 1176.41006 · doi:10.1016/j.jfa.2008.09.013 |
| [17] | Olevskii, A.; Ulanovskii, A., Functions with Disconnected Spectrum (2016), New York: American Mathematical Society, New York · Zbl 1350.42001 · doi:10.1090/ulect/065 |
| [18] | Ortega-Cerdà, J.; Seip, K., Fourier frames, Ann. Math., 155, 3, 789-806 (2002) · Zbl 1015.42023 · doi:10.2307/3062132 |
| [19] | Pavlov, B.S.: The basis property of a system of exponentials and the condition of Muckenhoupt. Dokl. Akad. Nauk SSSR 247(1), 37-40 (1979). (in Russian) |
| [20] | Seip, K., A simple construction of exponential bases in \(L^2\) of the union of several intervals, Proc. Edinburgh Math. Soc., 38, 1, 171-177 (1995) · Zbl 0826.30003 · doi:10.1017/S0013091500006295 |
| [21] | Young, RM, On complete biorthogonal system, Proc. Am. Math. Soc., 83, 3, 537-540 (1981) · Zbl 0484.42009 · doi:10.1090/S0002-9939-1981-0627686-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
