Equivalence of the trigonometric system and its perturbations in the spaces $L^p$ and $C$

A. M. Sedletskii

Sbornik. Mathematics, Tome 210 (2019) no. 4, pp. 606-624

Résumé

Let $B=B[-\pi,\pi]$ be any of the spaces $L^p(-\pi,\pi)$, $1\leq p<\infty$, $p\neq2$, and $C[-\pi,\pi]$, and let $B_a=B[-\pi+a,\pi+a]$, $a\in\mathbb R$. A number of necessary conditions and sufficient conditions for the ‘perturbed trigonometric system’ $e^{i(n+\alpha_n)t}$, $n\in\mathbb Z$, to be equivalent to the trigonometric system $e^{int}$, $n\in\mathbb Z$, in the space $B_a$ for any $a\in\mathbb R$ are obtained. In particular, it is shown that if $(\alpha_n)\in l^s$, where $1/s=|1/p-1/2|$, then this equivalence takes place, the exponent $s$ being sharp. This result is used to show that in $L^p(-\pi,\pi)$, $1<p<2$, there exist bases of exponentials which are not equivalent to the trigonometric basis. The machinery of Fourier multipliers is used in the proofs. Bibliography: 18 titles.

Détail

Keywords: equivalent systems of functions, basis
Mots-clés : Fourier multiplier.

Bibliographie

[1] M. I. Kadets, “The exact value of the Paley–Wiener constant”, Soviet Math. Dokl., 5 (1964), 559–561 | MR | Zbl

[2] A. M. Sedletskii, Klassy analiticheskikh preobrazovanii Fure i eksponentsialnye approksimatsii, Fizmatlit, M., 2005, 504 pp.

[3] W. O. Alexander, Jr., R. Redheffer, “The excess of sets of complex exponentials”, Duke Math. J., 34 (1967), 59–72 | DOI | MR | Zbl

[4] A. M. Sedletskii, “Biorthogonal expansions of functions in series of exponents on intervals of the real axis”, Russian Math. Surveys, 37:5 (1982), 57–108 | DOI | MR | Zbl

[5] E. I. Moiseev, “On the basis property of systems of sines and cosines”, Soviet Math. Dokl., 29 (1984), 296–300 | MR | Zbl

[6] G. G. Devdariani, Bazisnost nekotorykh spetsialnykh sistem sobstvennykh funktsii nesamosopryazhennykh differentsialnykh operatorov, Avtoref. diss. ... kand. fiz.-matem. nauk, MGU, M., 1985

[7] A. A. Yukhimenko, “Bases of exponentials in the weighted spaces $L^p(-\pi,\pi)$”, Moscow Univ. Math. Bull., 65:2 (2010), 69–71 | DOI | MR | Zbl

[8] A. M. Sedletskii, “Equivalence of the trigonometric system and its perturbations in $L^p(-\pi,\pi)$”, Dokl. Math., 94:1 (2016), 464–467 | DOI | MR | Zbl

[9] R. E. Edwards, Fourier series. A modern introduction, v. 2, Grad. Texts in Math., 85, 2nd ed., Springer-Verlag, New York–Berlin, 1982, xi+369 pp. | MR | MR | Zbl

[10] R. J. Duffin, J. J. Eachus, “Some notes on an expansion theorem of Paley and Wiener”, Bull. Amer. Math. Soc., 48:12 (1942), 850–855 | DOI | MR | Zbl

[11] A. M. Sedletskii, “Ekvivalentnye posledovatelnosti v nekotorykh prostranstvakh funktsii”, Izv. vuzov. Matem., 1973, no. 7, 85–91 | MR | Zbl

[12] A. Zygmund, Trigonometric series, v. I, II, 2nd ed., Cambridge Univ. Press, New York, 1959, xii+383 pp., vii+354 pp. | MR | MR | Zbl | Zbl

[13] E. Seneta, Regularly varying functions, Lecture Notes in Math., 508, Springer-Verlag, Berlin–New York, 1976, v+112 pp. | DOI | MR | MR | Zbl | Zbl

[14] A. M. Minkin, “Reflection of exponents and unconditional bases of exponentials”, St. Petersburg Math. J., 3:5 (1992), 1043–1068 | MR | Zbl

[15] A. M. Sedletskii, “Asymptotics of the zeros of degenerate hypergeometric functions”, Math. Notes, 82:2 (2007), 229–237 | DOI | DOI | MR | Zbl

[16] B. Ya. Levin, “O bazisakh pokazatelnykh funktsii v $L^2$”, Zap. matem. otd. fiz.-matem. f-ta Kharkovskogo un-ta i Kharkovskogo matem. o-va, ser. 4, 27 (1961), 39–48

[17] V. D. Golovin, “O biortogonalnykh razlozheniyakh v $L^2$ po lineinym kombinatsiyam pokazatelnykh funktsii”, Zap. matem. otd. fiz.-matem. f-ta Kharkovskogo un-ta i Kharkovskogo matem. o-va, ser. 4, 30 (1964), 18–29 | MR

[18] J. Ortega-Cerdà, K. Seip, “Fourier frames”, Ann. of Math. (2), 155:3 (2002), 789–806 | DOI | MR | Zbl