On the Cyclicity of Dilated Systems in Lattices: Multiplicative Sequences, Polynomials, Dirichlet-type Spaces and Algebras

[1] N. I. Ahiezer, Lectures on the theory of approximation, OGIZ, Moscow-Leningrad, 1947 (Russian), 323 pp (MR0025598). Search in Google Scholar

[2] A. Aleman, M. Hartz, J. McCarthy, and S. Richter, Free outer functions in complete Pick spaces, arXiv:2203.08179v1, 15 Mar 2022. Search in Google Scholar

[3] N. Arcozzi, R. Rochberg, E. Sawyer, Two variations on the Drury-Arveson space, in “Hilbert spaces of analytic functions” CRM Proc. Lecture Notes, 51, 41–58 (AMS, Providence RI), 2010. Search in Google Scholar

[4] L. Báez-Duarte, A strengthening of the Nyman–Beurling criterion for the Riemann hypothesis, Atti Acad. Naz. Lincei 14 (2003), 5–11. Search in Google Scholar

[5] L. Bergqvist, A note on cyclic polynomials in polydiscs, Anal. Math. Phys. (2018), 8: 197–211. Search in Google Scholar

[6] A. Beurling, On the completeness of {ψ (nt)} on L²(0, 1), a seminar talk 1945, The collected Works of Arne Beurling, Vol. 2: Harmonic Analysis, Contemp. Mathematicians, Birkhäuser, Boston, 1989, 378–380. Search in Google Scholar

[7] A. Beurling, A critical topology in harmonic analysis on semigroups, Acta Math. 112 (1964), 215–228. Search in Google Scholar

[8] H. Bohr,Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reien ∑ αnns \sum {{{{\alpha _n}} \over {{n^s}}}} , Nachr. Ges. Wiss. Göttingen. Math.-Phys. Kl., 1913, A9, 441–488. Search in Google Scholar

[9] A. Borichev, O. El-Fallah, A. Hanine, Cyclicity in weighted Bergman type spaces, J. Math. Pures Appl., (9)102:6(2014), 1041–1061. Search in Google Scholar

[10] A. Borichev, H. Hedenmalm, A. Volberg, Large Bergman spaces: invertibility, cyclicity, and subspaces of arbitrary index, J. Funct. Anal. 207:1 (2004), 111–160. Search in Google Scholar

[11] H. Dan and K. Guo, The periodic dilation completeness problem: cyclic vectors in the Hardy space over the infinite-dimensional polydisc, J. London Math. Soc., (2) 00 (2020), 1–34. Search in Google Scholar

[12] H. Dan, K. Guo, The Beurling–Wintner problem for step functions, arXiv:205.09779v3, 2021, 53pp. Search in Google Scholar

[13] H. Dan, K. Guo, J. Ni, Invariant subspaces of the Bergman spaces in infinitely many variables, arXiv:2103.04145v2, 2021. Search in Google Scholar

[14] O. El-Fallah, K. Kellay, J. Mashreghi, Th. Ransford, A primer on the Dirichlet spaces, Cambridge Univ. Press, Cambridge, 2014. Search in Google Scholar

[15] O. El-Fallah, K. Kellay, Th. Ransford, Cyclicity in the Dirichlet spaces, Ark. Mat. 44:1, 2006, 61–86. Search in Google Scholar

[16] O. El-Fallah, N. Nikolski, and M. Zarrabi, Resolvent estimates in Beurling–Sobolev algebras, Algebra i Analiz, 6 (1998), 1–80 (Russian); English transl.: St. Petersburg Math. J., 6 (1999), 1–69. Search in Google Scholar

[17] Q. Fang, Multipliers of Drury–Arveson space: a survey, pp. 99-116 in: Oper. Theory Adv. Appl., vol. 272, Birkhäuser, 2019. Search in Google Scholar

[18] Ph. Hartman, Multiplicative sequences and Töplerian (L²)-bases, Duke Math. J., 14 (1947), 755–767. Search in Google Scholar

[19] H. Hedenmalm, P. Lindquist, and K. Seip, A Hilbert space of Dirichlet series and sytems of dilated functions in L²(0, 1), Duke Math. J., 86 (1997), 1–37. Search in Google Scholar

[20] H. Hedenmalm, P. Lindquist, and K. Seip, Addendum to “A Hilbert space of Dirichlet series and sytems of dilated functions in L²(0, 1)”, Duke Math. J., 99 (1999), 175–178. Search in Google Scholar

[21] D. Hilbert, Wesen und Ziele einer Analysis der unendlich vielen unabhängigen Variablen, Rend. Cir. Mat. Palermo 27 (1909). Search in Google Scholar

[22] K. Kellay, F. LeManach, M. Zarrabi, Cyclicity in Dirichlet type spaces, Contemp. Math., v.743, 181–193, Amer. Math. Soc., Providence, RI, 2020. Search in Google Scholar

[23] E. Kerlin, A. Lambert, Strictly cyclic shifts on ℓ_p, Acta Sci. Math. (Szeged), 35 (1973), 87–94. Search in Google Scholar

[24] G. Knese, L. Kosinski, T. Ransford, A. Sola, Cyclic polynomials in anisotropic Dirichlet spaces, J. Anal. Math., 138:1 (2019), 23–47. Search in Google Scholar

[25] L. Kosinski, D. Vavitsas, Cyclic polynomials in Dirichlet-type spaces in the unit ball of ℂ², arXiv:2212.12013v1, 19pp, 2022. Search in Google Scholar

[26] V. Ya. Kozlov, On the completeness of systems of functions {φ(nx)} in the space L²(0, 2π)(Russian), Doklady Akad. Nauk SSSR, 61 (1948), 977–980. Search in Google Scholar

[27] V. Ya. Kozlov, On the completeness of the system of functions {φ(nx)} in the space of odd functions from L²(0, 2π)(Russian), Doklady Akad. Nauk SSSR, 62 (1948), 13–16. Search in Google Scholar

[28] V. Ya. Kozlov, On the completeness of a system of functions of type {φ(nx)} in the space L²(Russian), Doklady Akad. Nauk SSSR, 73 (1950), 441–444. Search in Google Scholar

[29] F. Le Manach, Sur l’approximation et la complétude des translatés dans les espaces de fonctions, Thèse de Doctorat, 108 pp. Univ. Bordeaux, 2018 (HAL https://Imf.alea.ovh/Recherche/These/memoire.pdf). Search in Google Scholar

[30] F. Le Manach, Cyclicity in weighted ℓ^p spaces, Preprint, https://HAL.archives-ouvertes.fr/hal-01483933v1. Search in Google Scholar

[31] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, vol. I & II, Springer-Verlag, Berlin etc., 1979. Search in Google Scholar

[32] R. Merris, Combinatorics, J.Wiley, 2003. Search in Google Scholar

[33] B. Mityagin, Systems of dilated functions: completeness, minimality, basisness, Funct. Anal. Appl., 51:3 (2017), 236–239; see also arXiv:1612.06791v (20 Dec 2016). Search in Google Scholar

[34] Multiplicative function, https://en.wikipedia.org/wiki/Multiplicative_function. Search in Google Scholar

[35] J. Neuwirth, J. Ginsberg, and D. Newman, Approxiamtion by f(kx), J. Funct. Anal., 5 (1970), 194–203. Search in Google Scholar

[36] N. Nikolski, Basicity and unicellularity of weighted shift operators, Izvestia Akad. Nauk SSSR, Ser. Mat. 32 (1968), 1123–1137 (Russian); Engl. transl.: MatH. URSS Izv. 2 (1968), 1077–1090. Search in Google Scholar

[37] N. Nikolski, Spectral synthesis for a shift operator and zeroes in certain classes of analytic functions smooth up to the boundary, Dokl. Akad. Nauk SSSR, 190 (1970), 780–783 (Russian); Engl. transl.: Soviet Math. Dokl. 11 (1970), 206–209. Search in Google Scholar

[38] N. Nikolski, Selected problems of weighted approximation and spectral analysis, Trudy Math. Inst. Steklova, vol. 120, Moscow (Russian); English transl.: Proc. Steklov Math. Inst., No 120 (1974), AMS, Providence, 1976. Search in Google Scholar

[39] N. Nikolski, Two problems on the spectral synthesis, Zapiski Nauchn. Seminarov LOMI, 81, 139–141; reprinted in: J. Soviet Math. 26 (1984), 2185–2186; reprinted in Lect. Notes Math., Springer-Verlag, vol. 1043 (1984), 378–381. Search in Google Scholar

[40] N. Nikolski, Treatise on the shift operator, Springer-Verlag, Berlin etc., 1986 (Russian original: Lekzii ob operatore sdviga, “Nauka”, Moskva, 1980) Search in Google Scholar

[41] N. Nikolski, In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc,Ann. Inst. Fourier, 62:5 (2012), 1601–1626. Search in Google Scholar

[42] N. Nikolski, Binomials whose dilations generate H²(D), Algebra and Analysis (St. Petersburg), 29:6 (2017), 159–177. Search in Google Scholar

[43] N. Nikolski, Addendum and Corrigendum to the paper “In a shadow of the RH: cyclic vectors of Hardy spaces on the Hilbert multidisc, Ann. Inst. Fourier, 62:5 (2012), 1601–1626”, Ann. Inst. Fourier (Grenoble), 68:2 (2018), 563–567. Search in Google Scholar

[44] N. Nikolski, The current state of the dilation completeness problem, A PPT talk in the MSU analysis seminar 2018, and in the KCL (London) analysis seminar 2017. Search in Google Scholar

[45] N. Nikolski, Hardy Spaces, Cambridge Univ. Press, 2019. Search in Google Scholar

[46] B. Nyman, On the one-dimensional translation group and semi-group in certain function spaces, Thesis, Uppsala Univ., 1950. Search in Google Scholar

[47] J. Ortega, J. Fabrega, Multipliers in Hardy–Sobolev spaces, Integral Eq. Operator Theory, 55(4) (2006), 535–560. Search in Google Scholar

[48] https://en.wikipedia.org/wiki/Pascal%27s_triangle Search in Google Scholar

[49] G. Polya and G. Szegő, Problems and theorems in Analysis, Vol. I & II, Springer, 1976 (for 3rd English edition); for original German edition “Aufgaben und Lehrsätze aus der Analysis”, Band I & II, Springer, Berlin, 1925). Search in Google Scholar

[50] H. Queffélec, Espaces de séries de Dirichlet et leurs opérateurs de composition, Annales math. Blaise Pascal, 22 N S2 (2015), 267–314. Search in Google Scholar

[51] S. Richter, C. Sundberg, Cyclic vectors in the Drury-Arveson space, ESI Workshop on “Operator related function theory”, 2012 (web.math.utk.edu/_~richter/talk/), https://web.math.utk.edu/~richter/talk/Richter_Cyclic_Drury_Arveson2012.pdf. Search in Google Scholar

[52] S. Richter, J. Sunkes, Hankel operators, invariant subspaces, and cyclic vectors in the Drury-Arveson space, Proc. Amer. Math. Soc., 114:6 (2016), 2575–2586. Search in Google Scholar

[53] N. P. Romanov, Hilbert space and the theory of numbers, Izvestia Acad. Nauk SSSR, 10 (1946), 3–34 (Russian); MR0016390. Search in Google Scholar

[54] W. Rudin, Function theory in polydiscs, W. A. Benjamin, Inc, N.Y.-Amsterdam, 1969. Search in Google Scholar

[55] W. Rudin, Function theory in the unit ball of ℂⁿ, Springer-Verlag, Berlin-N.Y., 1980. Search in Google Scholar

[56] O. Shalit, Operator theory and function theory in Drury-Arveson space and its quotients, in “Operator Theory, Adv. and Appl.” (ed.: D. Alpay), pp. 1–50, Springer, Basel, 2014. Search in Google Scholar

[57] A. Shields, Weighted shift operators and analytic function theory, pp. 49–128 in: Topics in operator theory, ed. C. Pearcy, Math. Surveys No 13, Amer. Math. Soc., 1974, Providence. Search in Google Scholar

[58] https://en.wikipedia.org/wiki/Vandermonde%27s_identity Search in Google Scholar

[59] J. Wermer, On a class of normed rings, Ark. Mat. 2:6 (1954), 537–551. Search in Google Scholar

[60] A. Wintner, Diophantine approximation and Hilbert’s space, Amer. J. Math., 66 (1944), 564–578. Search in Google Scholar

[61] K. Zhu, Spaces of holomorphic functions in the unit ball, Springer, N.Y., 2005. Search in Google Scholar