zbMATH — the first resource for mathematics

On the stability of the uniform minimality of a set of exponentials. (English. Russian original) Zbl 1162.30006
J. Math. Sci., New York 155, No. 1, 170-182 (2008); translation from Sovrem. Mat., Fundam. Napravl. 25, 165-177 (2007).
Summary: Some conditions on the closeness of sequences \((\lambda_n)\) and \((\mu_n)\) are given which ensure that the corresponding systems of complex exponentials \((\exp(i\lambda_n t))\) and \((\exp(i\mu _n t))\) are simultaneously uniformly minimal in \(L^p(-\pi, \pi)\), \(1\leq p < \infty\), and \(C[-\pi, \pi]\).
30B60 Completeness problems, closure of a system of functions of one complex variable
Full Text: DOI
[1] W. O. Alexander and R. Redheffer, ”The excess of sets of complex exponentials,” Duke Math. J., 34, 59–72 (1967). · Zbl 0147.33902
[2] R. Edwards, A Modern Treatment of Fourier Series, Vol. 2 [Russian translation], Mir, Moscow, (1985). · Zbl 0662.42001
[3] J. Elsner, ”Zulässige Abänderungen von Exponentialsystemen im L p (, A),” Math. Z., 120, 211–220 (1971). · Zbl 0208.15101
[4] V. A. Il’in, ”Necessary and sufficient conditions for a system to be a basis in L p and for the equiconvergence with a trigonometric series of spectral decompositions with respect to a system of exponentials,” Dokl. Akad. Nauk SSSR, 273, No. 4, 789–793 (1983).
[5] S. G. Krein, Functional Analysis [in Russian], Nauka, Moscow (1972).
[6] P. Kukis, An Introduction to the theory of spaces H p [Russian translation], Mir, Moscow, (1984).
[7] V. D. Milman, ”Geometric theory of Banach spaces,” Usp. Mat. Nauk., 25, No. 3, 113–174 (1970).
[8] R. Redheffer, ”Completeness of sets of complex exponentials,” Adv. Math., 24, 1–62 (1977). · Zbl 0358.42007
[9] A. M. Sedletskii, ”The excess of sets of exponentials,” Mat. Zametki, 22, No. 6, 803–814 (1977).
[10] A. M. Sedletskii, ”The excess of sets of exponentials,” Izv. Akad. Nauk SSSR, Ser. Mat., 44, No. 1, 203–218 (1980).
[11] A. M. Sedletskii, ”The excess of nearby sets of exponentials in L p ,” Sib. Mat. Zh., 24, No. 4, 164–175 (1983).
[12] A. M. Sedletskii, ”On purely imaginary perturbations of the exponents {\(\lambda\)} n for the system {ie182-01},” Sib. Mat. Zh., 26, No. 4, 151–158 (1985).
[13] A. M. Sedletskii, ”Approximative properties of systems of exponentials in L p (a, b),” Differ. Uravn., 31, No. 10, 1639–1645 (1995).
[14] A. M. Sedletskii, ”On entire functions of the S. N. Bernstein class which are not images under the Fourier-Stieltjes transform,” Mat. Zametki, 61, No. 3, 367–380 (1997).
[15] A. M. Sedletskii, ”The stability of sets of finite Fourier transforms,” in: Integral Transformations and Special Functions, Information bulletin. CC RAS Press, 1, No. 2, (1997), pp. 17–19.
[16] E. C. Titchmarsh, ”The zeros of certain integral functions,” Proc. London Math. Soc. Ser. 2., 25, 283–302 (1926). · JFM 52.0334.03
[17] R. Young, ”On perturbing bases of complex exponentials in L 2(, {\(\pi\)}),” Proc. Amer. Math. Soc., 53, 137–140 (1975). · Zbl 0319.46015
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.