Geodesic
Generalized Dirichlet classes in a half-plane and their application to approximations
Sbornik. Mathematics, Tome 206 (2015) no. 1, pp. 135-160 Cet article a éte moissonné depuis la source Math-Net.Ru

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We introduce generalized Dirichlet classes of analytic functions in a disc and a half-plane. We establish a relationship between these classes and their zero sets. A precise sufficient condition for a zero subset of a generalized Dirichlet class in a half-plane is obtained. Using this condition, we prove a necessary condition (which is also precise) for a system of exponential functions to be complete in the space $L^2$ on a half-line with regularly varying weight of order $\alpha\in[-1,0]$. Bibliography: 18 titles.
Keywords: slowly varying function, generalized Bergman and Dirichlet classes, zero set, completeness of a system of exponentials.
Mots-clés : Laplace transform 
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A. M. Sedletskii. Generalized Dirichlet classes in a half-plane and their application to approximations. Sbornik. Mathematics, Tome 206 (2015) no. 1, pp. 135-160. http://geodesic.mathdoc.fr/item/SM_2015_206_1_a8/

[1] O. Szász, “Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen”, Math. Ann., 77:4 (1916), 482–496 | DOI | MR | Zbl

[2] L. Schwartz, Étude des sommes d'exponentielles, Actualités Sci. Ind., 2ième éd., Hermann, Paris, 1959, 151 pp. | MR | Zbl

[3] L. Carleson, “On the zeros of functions with bounded Dirichlet integral”, Math. Z., 56:3 (1952), 289–295 | DOI | MR | Zbl

[4] H. S. Shapiro, A. L. Shields, “On the zeros of functions with finite Dirichlet integral and some related function spaces”, Math. Z., 80:1 (1962), 217–229 | DOI | MR | Zbl

[5] I. O. Krasnobaev, “Neobkhodimoe uslovie polnoty sistemy eksponent v prostranstve kvadratichno integriruemykh funktsii so stepennym vesom”, Anal. Math., 38:3 (2012), 187–202 | DOI | MR | Zbl

[6] A. Nagel, W. Rudin, J. H. Shapiro, “Tangential boundary behavior of functions in Dirichlet-type spaces”, Ann. of Math. (2), 116:2 (1982), 331–360 | DOI | MR | Zbl

[7] A. M. Sedletskii, Klassy analiticheskikh preobrazovanii Fure i eksponentsialnye approksimatsii, Fizmatlit, M., 2005, 503 pp.

[8] B. N. Khabibullin, “Polnota sistem eksponent v prostranstvakh funktsii na luche i kharakteristika Korenblyuma–Seipa”, Ufimsk. matem. zhurn., 1:1 (2009), 77–83 | Zbl

[9] A. M. Sedletskii, “Approximation of Müntz–Szász type in weighted spaces”, Sb. Math., 204:7 (2013), 1028–1055 | DOI | DOI | MR | Zbl

[10] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular variation, Encyclopedia Math. Appl., 27, Cambridge Univ. Press, Cambridge, 1987, xx+491 pp. | DOI | MR | Zbl

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

[12] G. Doetsch, Handbuch der Laplace-Transformation, v. I, Die theoretischen Grundlagen der Laplace-Transformation, Verlag Birkhäuser, Basel, 1950, 581 pp. | MR | Zbl

[13] W. Feller, An introduction to probability theory and its applications, v. II, 2nd ed., John Wiley Sons, Inc., New York–London–Sydney, 1971, xxiv+669 pp. | MR | MR | Zbl | Zbl

[14] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman spaces, Grad. Texts in Math., 199, Springer-Verlag, New York, 2000, x+286 pp. | DOI | MR | Zbl

[15] L. Biryukov, “Some theorems on integrability of Laplace transforms and their applications”, Integral Transforms Spec. Funct., 18:7-8 (2007), 459–469 | DOI | MR | Zbl

[16] A. M. Sedletskii, “Müntz–Szász type approximation in direct products of spaces”, Izv. Math., 70:5 (2006), 1031–1050 | DOI | DOI | MR | Zbl

[17] A. R. Siegel, “On the Müntz–Szász theorem for $C[0,1]$”, Proc. Amer. Math. Soc., 36:1 (1972), 161–166 | DOI | MR | Zbl

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