## Completeness of the trigonometric system for the classes $$\varphi (L)$$.(English. Russian original)Zbl 1136.46026

Math. Notes 81, No. 5, 632-637 (2007); translation from Mat. Zametki 81, No. 5, 707-712 (2007).
Summary: We obtain a necessary and sufficient condition for the completeness of the trigonometric system with gaps for the classes $$\varphi (L)$$.

### MSC:

 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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### References:

 [1] P. L. Ul’anov, ”The representation of functions by series and the classes (L),” Uspekhi Mat. Nauk 27(2), 3–52 (1972) [Russian Math. Surveys 27 (2), 1–54 (1972)]. [2] V. I. Ivanov, ”Representation of functions by series in metric symmetric spaces without linear functionals,” in Proc. Steklov Inst. Math. (Nauka, Moscow, 1989), Vol. 189, pp. 34–77 [in Russian]. [3] V. I. Filippov, ”Linear continuous functionals and representation of functions by series in the spaces E ,” Anal. Math. 27(4), 239–260 (2001). · Zbl 1003.46020 [4] A. A. Talalyan, ”Representation of functions of classes L p[0, 1], 0 < p < 1, by orthogonal series,” Acta Math. Academ. Sci. Hungar. 21(1–2), 1–9 (1970). · Zbl 0208.09202 [5] K. de Leeuw, ”The failure of spectral analysis in L p for 0 < p < 1,” Bull. Amer. Math. Soc. 82(1), 111–114 (1976). · Zbl 0322.43009 [6] J. H. Shapiro, ”Subspaces of L p(G) spanned by characters: 0 < p < 1,” Israel J. Math. 29(2–3), 248–264 (1978). · Zbl 0382.46015 [7] A. B. Aleksandrov, ”Essays on nonlocally convex Hardy classes,” in Complex Analysis and Spectral Theory, in Lecture Notes in Math. (Springer-Verlag, Berlin, 1981), Vol. 864, pp. 1–89. [8] V. I. Ivanov and V. A. Yudin, ”On the trigonometric system in L p, 0 < p < 1,” Mat. Zametki 28(6), 859–868 (1980). · Zbl 0455.42016 [9] V. I. Ivanov, ”Representation of measurable functions by multiple trigonometric series,” in Proc. Steklov Inst. Math. (Nauka, Moscow, 1983), Vol. 164, pp. 100–123 [in Russian]. · Zbl 0565.42006 [10] D. Ya. Spivakovskaya, ”On the trigonometric system in the metric spaces L {$$\Psi$$},” Vestnik Dnepropetrovsk. Nats. Univ., No. 6, 101–115 (2001). [11] A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 1959, 1960; Mir, Moscow, 1965), Vols. 1, 2. [12] M. A. Krasnosel’skii and Ya. B. Rutitskii, Convex Functions and Orlicz Spaces (Fizmatgiz, Moscow, 1958) [in Russian]. [13] R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions (Kluwer Acad. Publ., Dordrecht, 2004).
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