Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Subspaces of de Branges spaces with prescribed growth

Authors: A. Baranov and H. Woracek
Original publication: Algebra i Analiz, tom 18 (2006), nomer 5.
Journal: St. Petersburg Math. J. 18 (2007), 699-716
MSC (2000): Primary 46E20; Secondary 46E22, 30D15
Published electronically: August 9, 2007
MathSciNet review: 2301039
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The growth properties of de Branges spaces and their subspaces are studied. It is shown that, for each given pair of growth functions $ \lambda(r)=O(r)$ and $ \lambda_1=o(\lambda)$, there exist de Branges spaces of growth $ \lambda$ that have a de Branges subspace of growth $ \lambda_1$. This phenomenon cannot occur for a class of de Branges spaces that, in a certain sense, behave regularly along the real axis.

References [Enhancements On Off] (What's this?)

  • [ADRS] D. Alpay, A. Dijksma, J. Rovnyak, and H. de Snoo, Schur functions, operator colligations, and reproducing kernel Pontryagin spaces, Oper. Theory Adv. Appl., vol. 96, Birkhäuser Verlag, Basel, 1997. MR 1465432 (2000a:47024)
  • [B1] A. D. Baranov, The Bernstein inequality in the de Branges spaces and embedding theorems, Trudy S.-Peterburg. Mat. Obshch. 9 (2001), 23-53; English transl., Amer. Math. Soc. Transl. Ser. 2, vol. 209, Amer. Math. Soc., Providence, RI, 2003, pp. 21-49. MR 2018371 (2004m:30043)
  • [B2] -, Polynomials in the de Branges spaces of entire functions, Ark. Mat. 44 (2006), no. 1, 16-38. MR 2237209
  • [BP1] C. Berg and H. Pedersen, On the order and type of the entire functions associated with an indeterminate Hamburger moment problem, Ark. Mat. 32 (1994), 1-11. MR 1277917 (95i:30027)
  • [BP2] -, Nevanlinna matrices of entire functions, Math. Nachr. 171 (1995), 29-52. MR 1316350 (96a:47028)
  • [BP3] -, Logarithmic order and type of indeterminate moment problems (with an appendix by W. Hayman), Difference Equations, Special Functions, and Applications (Munich, July 25-30, 2005) (to appear).
  • [dB] L. de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1968. MR 0229011 (37:4590)
  • [DK1] H. Dym and H. McKean, Application of de Branges spaces of integral functions to the prediction of stationary Gaussian processes, Illinois J. Math. 14 (1970), 299-343. MR 0260013 (41:4642)
  • [DK2] -, Gaussian processes, function theory, and the inverse spectral problem, Probab. Math. Statist., vol. 31, Acad. Press, New York-London, 1976. MR 0448523 (56:6829)
  • [GM] L. Golinskii and I. Mikhailova, Hilbert spaces of entire functions as a $ J$-theory subject, Topics in Interpolation Theory (Leipzig, 1994), Oper. Theory Adv. Appl., vol. 95, Birkhäuser, Basel, 1997, pp. 205-251. MR 1473258 (98m:46031)
  • [KaK1] I. S. Kac and M. G. Krein, $ R$-functions - analytic functions mapping the upper halfplane into itself, F. V. Atkinson. Discrete and Continuous Boundary Problems, ``Mir,'' Moscow, 1968, Supplement I to the Russian ed., pp. 629-647; English transl., Amer. Math. Soc. Transl. (2), vol. 103, Amer. Math. Soc., Providence, RI, 1974, pp. 1-18. MR 0243149 (39:4473)
  • [KaK2] -, On the spectral functions of the string, F. V. Atkinson. Discrete and Continuous Boundary Problems, ``Mir,'' Moscow, 1968, Supplement II to the Russian ed., pp. 648-737; English transl., Amer. Math. Soc. Transl. (2), vol. 103, Amer. Math. Soc., Providence, RI, 1974, pp. 19-102. MR 0243149 (39:4473)
  • [KWW1] M. Kaltenbäck, H. Winkler, and H. Woracek, Singularities of generalized strings, Operator Theory and Indefinite Inner Product Spaces, Oper. Theory Adv. Appl., vol. 163, Birkhäuser, Basel, 2006, pp. 191-248. MR 2215864 (2007f:47040)
  • [KWW2] -, De Branges spaces of entire functions symmetric about the origin, Integral Equations Operator Theory 56 (2006), no. 4, 483-509. MR 2284712
  • [KW1] M. Kaltenbäck and H. Woracek, Pontryagin spaces of entire functions. I, Integral Equations Operator Theory 33 (1999), 34-97. MR 1664343 (2000a:46039)
  • [KW2] -, Pólya class theory for Hermite-Biehler functions of finite order, J. London Math. Soc. (2) 68 (2003), no. 2, 338-354. MR 1994686 (2004e:30042)
  • [KW3] -, De Branges spaces of exponential type: General theory of growth, Acta Sci. Math. (Szeged) 71 (2005), no. 1-2, 231-284. MR 2160366 (2006c:30031)
  • [K1] M. G. Krein, A contribution to the theory of entire functions of exponential type, Izv. Akad. Nauk SSSR Ser. Mat. 11 (1947), no. 4, 309-326. (Russian) MR 0022252 (9:179e)
  • [K2] -, On the indeterminate case of the Sturm-Liouville boundary problem in the interval $ (0,\infty )$, Izv. Akad. Nauk SSSR Ser. Mat. 16 (1952), no. 4, 293-324. (Russian) MR 0052004 (14:558g)
  • [KL] M. G. Krein and H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume $ \Pi_\kappa$ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187-236. MR 0461188 (57:1173)
  • [LG] P. Lelong and L. Gruman, Entire functions of several complex variables, Grundlehren Math. Wiss., Bd. 282, Springer-Verlag, Berlin, 1986. MR 0837659 (87j:32001)
  • [L] B. Levin, Nullstellenverteilung ganzer Funktionen, Math. Lehrbücher Monogr. II. Abt. Math. Monogr., Bd. 14, Akademie-Verlag, Berlin, 1962. MR 0150301 (27:302)
  • [Li] X.-J. Li, Hilbert spaces of entire functions and polynomials orthogonal on the unit circle, Methods Appl. Anal. 1 (1994), no. 1, 25-43. MR 1260381 (95a:30004)
  • [OS] J. Ortega-Cerdà and K. Seip, Fourier frames, Ann. of Math. (2) 155 (2002), 789-806. MR 1923965 (2003k:42055)
  • [P] G. Pick, Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1916), 7-23. MR 1511844
  • [PW] V. Pivovarchik and H. Woracek, Shifted Hermite-Biehler functions and their applications, Integral Equations Operator Theory 57 (2007), no. 1, 101-126. MR 2294277
  • [Re] C. Remling, Schrödinger operators and de Branges spaces, J. Funct. Anal. 196 (2002), no. 2, 323-394. MR 1943095 (2003j:47055)
  • [RosR] M. Rosenblum and J. Rovnyak, Topics in Hardy classes and univalent functions, Birkhäuser Verlag, Basel, 1994. MR 1307384 (97a:30047)
  • [RR] J. Rovnyak and V. Rovnyak, Sonine spaces of entire functions, J. Math. Anal. Appl. 27 (1969), 68-100. MR 0243333 (39:4655)
  • [Ru] L. A. Rubel, Entire and meromorphic functions, Springer-Verlag, New York, 1996. MR 1383095 (97c:30001)
  • [Wi] H. Winkler, On transformations of canonical systems, Operator Theory and Boundary Eigenvalue Problems (Vienna, 1993), Oper. Theory Adv. Appl., vol. 80, Birkhäuser, Basel, 1995, pp. 276-288. MR 1362114 (96m:34155)
  • [W] H. Woracek, De Branges spaces of entire functions closed under forming difference quotients, Integral Equations Operator Theory 37 (2000), no. 2, 238-249. MR 1769812 (2001g:46058)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 46E20, 46E22, 30D15

Retrieve articles in all journals with MSC (2000): 46E20, 46E22, 30D15

Additional Information

A. Baranov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospect 28, Staryĭ Peterhof, St. Petersburg 198504, Russia

H. Woracek
Affiliation: Institute of Mathematics, Royal Institute of Technology (KTH), SE-100, 44, Stockholm, Sweden
Address at time of publication: Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8-10/101, A-1040, Wien, Austria

Keywords: de Branges space, growth function, de Branges subspace
Received by editor(s): April 22, 2006
Published electronically: August 9, 2007
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society