×

\(L^2\)-Müntz spaces as model spaces. (English) Zbl 1462.46025

Summary: We emphasize a bridge between two areas of function theory: Hilbertian Müntz spaces and model spaces of the Hardy space of the right half plane. We give miscellaneous applications of this viewpoint to Hilbertian Müntz spaces.

MSC:

46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
30H10 Hardy spaces
30B10 Power series (including lacunary series) in one complex variable
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] AlAlam, I., Habib, G., Lefèvre, P., Maalouf, F.: Essential norms of Volterra and Cesàro operators on Müntz spaces. To appear in Colloq. Math · Zbl 1502.47049
[2] Almira, J.M.: Müntz type theorems I. Surv. Approx. Theory 3, 152-194 (2007) · Zbl 1181.41001
[3] Baranov, A.: Completeness and Riesz Bases of Reproducing Kernels in Model Subspaces, pp. 1-34. IMRN(2006) · Zbl 1118.47007
[4] Borwein, P., Erdélyi, T.: The full Müntz theorem in \[C[0,1]C\][0,1] and \[L^1[0,1]\] L1[0,1]. J. Lond. Math. Soc. (2) 54(1), 102-110 (1996) · Zbl 0854.41015 · doi:10.1112/jlms/54.1.102
[5] Borwein, P., Erdélyi, T., Zhang, J.: Müntz systems and orthogonal Müntz Legendre polynomials. Trans. A.M.S. 342(2), 523-542 (1994) · Zbl 0799.41015
[6] Chalendar, I., Fricain, E., Timotin, D.: Embeddings theorems for Müntz spaces. Ann. l’Inst. Fourier 61, 2291-2311 (2011) · Zbl 1255.46013 · doi:10.5802/aif.2674
[7] Feinerman, R.P., Newman, D.J.: Polynomial Approximation. Williams and Wilkins, Baltimore (1974) · Zbl 0309.41006
[8] Fricain, E., Mashreghi, J.: The Theory of \[{\cal{H}}(b)H\](b) Spaces, vol. 1. New Mathematical Monographs: 20. Cambridge University Press (2016) · Zbl 1364.30002
[9] Gaillard, L., Lefèvre, P.: Lacunary Müntz spaces: isomorphisms and Carleson embeddings, submitted · Zbl 1412.30005
[10] Gurariy, V., Macaev, V.: Lacunary power sequences in spaces \[CC\] and \[L^p\] Lp, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 30, 3-14 (1966)
[11] Garcia S., Mashreghi J., Ross W.: Introduction to model spaces and their operators. Cambridge Studies in Advanced Mathematics, vol. 148. Cambridge University Press (2016) · Zbl 1361.30001
[12] Gurariy, V., Lusky, W.: Geometry of Müntz spaces and related questions. In: Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, New York (2005) · Zbl 1094.46003
[13] Hruschev, S., Nikolski, N., Pavlov, B.: Unconditional bases of exponentials and of reproducing kernels, Complex analysis and spectral theory. In: Lecture Notes in Mathematics 864, pp. 214-335. Berlin-Heidelberg-New York. Springer (1981)
[14] Kahane, J.P.: Some random series of functions. Cambridge University Press, Cambridge (1985) · Zbl 0571.60002
[15] Müntz, C.: Über den Approximationssatz von Weierstrass, pp. 303-312. H. A. Schwarz’s Festschrift, Berlin (1914) · JFM 45.0633.02
[16] Nikolski, N.: Treatise on the Shift operator, Grundlehren der Mathematischen Wissenschaften, vol. 273. Springer, Berlin (1986) · doi:10.1007/978-3-642-70151-1
[17] Nikolski N.: Operators, functions, and systems: an easy reading, vol. 2. Model operators and systems. Mathematical Surveys and Monographs, vol. 93. American Mathematical Society, USA (2002) · Zbl 1007.47002
[18] Noor, S.W., Timotin, D.: Embeddings of Müntz spaces: the Hilbertian case. Proc. Am. Math. Soc. 141, 2009-2023 (2013) · Zbl 1282.46026 · doi:10.1090/S0002-9939-2012-11681-8
[19] Gorkin, P., McCarthy, J., Pott, S., Witt, B.: Thin sequences and the Gram-Schmidt matrix. Arch. Math. 103, 93-99 (2014) · Zbl 1294.47032 · doi:10.1007/s00013-014-0667-8
[20] Rudin W.: Real and complex analysis, Third edition. McGraw-Hill Book Co., New York (1987) · Zbl 0925.00005
[21] Szász, O.: Über die approximation stetiger Funktionen durch lineare aggregate von Potenzen. Math. Ann. 77, 482-496 (1916) · JFM 46.0419.03 · doi:10.1007/BF01456964
[22] Volberg, S.: Two remarks concerning the theorem of S. Axler, S-Y.A Chang and D. Sarason, J.O.T. 7 no. 2, pp. 209-218 (1982)
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.