Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

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



Beurling-Malliavin multiplier theorem: The seventh proof

Authors: J. Mashreghi, F. L. Nazarov and V. P. Havin
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 17 (2005), nomer 5.
Journal: St. Petersburg Math. J. 17 (2006), 699-744
MSC (2000): Primary 42A50, 30D55
Published electronically: July 20, 2006
MathSciNet review: 2241422
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present a new proof of the Beurling-Malliavin theorem, often called the ``multiplier theorem'', concerning the existence of a real-valued function on $ \mathbb{R}$ with spectrum in a given (small) interval and with a given small majorant of the modulus. This proof pertains entirely to real analysis. It only involves elementary facts about the Hilbert transformation; neither complex variable methods nor potential theory is exploited. The heart of the proof is Theorem 2, which treats preservation of the Lipschitz property under the Hilbert transformation. We also include a short survey of earlier proofs of the Beurling--Malliavin theorem and its generalizations to model (coinvariant) subspaces of the Hardy space $ H^2(\mathbb{R})$.

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

  • [BAR] A. D. Baranov, Polynomials in the de Branges space of entire functions, Ark. Mat. (to appear).
  • [BBH] A. D. Baranov, A. A. Borichev, and V. P. Havin, Admissible majorants for meromorphic functions with fixed poles (in preparation).
  • [BE] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1948), 17 pp. MR 0027954 (10:381e)
  • [BH] A. D. Baranov and V. P. Havin, Admissible majorants for model subspaces and arguments of inner functions (in preparation).
  • [BK] K. Barbey and H. König, Abstract analytic function theory and Hardy algebras, Lecture Notes in Math., vol. 593, Springer-Verlag, Berlin-New York, 1977. MR 0442690 (56:1071)
  • [BM1] A. Beurling and P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (1962), 291-309. MR 0147848 (26:5361)
  • [BM2] -, On the closure of characters and the zeros of entire functions, Acta Math. 118 (1967), 79-93. MR 0209758 (35:654)
  • [BeH] Yu. S. Belov and V. P. Khavin, On a theorem of I. I. Privalov on the Hilbert transform of Lipschitz functions, Mat. Fiz. Anal. Geom. 11 (2004), no. 4, 380-407. (Russian) MR 2114001 (2005k:26006)
  • [Bu] M. A. Bulgakov, White guard. Novels. The Master and Margarita, Lenizdat, Leningrad, 1989. (Russian)
  • [DK] K. M. D'yakonov, Moduli and arguments of analytic functions from subspaces in $ H^p$ that are invariant under the backward shift operator, Sibirsk. Mat. Zh. 31 (1990), no. 6, 64-79; English transl., Siberian Math. J. 31 (1990), no. 6, 926-939 (1991). MR 1097956 (92f:30049)
  • [D] P. Duren, Theory of $ H^p$ spaces, Pure and Appl. Math., vol. 38, Acad. Press, New York-London, 1970. MR 0268655 (42:3552)
  • [DeB] L. de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1968. MR 0229011 (37:4590)
  • [DyM] H. Dym and H. P. McKean, Gaussian processes, function theory, and the inverse spectral problem, Probab. Math. Statist., vol. 31, Acad. Press, New York-London, 1976. MR 0448523 (56:6829)
  • [GLO] A. A. Gol'dberg, B. Ya. Levin, and I. V. Ostrovskii, Entire and meromorphic functions, Complex Analysis. One Variable, 1, Itogi Nauki i Tekhniki Sovrem. Probl. Mat. Fund. Naprav., vol. 85, VINITI, Moscow, 1991, pp. 5-185; English transl., Encyclopaedia Math. Sci., vol. 85, Springer, Berlin, 1997, pp. 1-193. MR 1155417 (93a:30031); MR 1464198
  • [Ga] J. B. Garnett, Bounded analytic functions, Pure and Appl. Math., vol. 96, Acad. Press, Inc., New York-London, 1981. MR 0628971 (83g:30037)
  • [Gam] T. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1969. MR 0410387 (53:14137)
  • [HJ] V. Havin and B. Jöricke, The uncertainty principle in harmonic analysis, Ergeb. Math. Grenzgeb. (3), vol. 28, Springer-Verlag, Berlin, 1994. MR 1303780 (96c:42001)
  • [HLP] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1988. MR 0944909 (89d:26016)
  • [HMI] V. Havin and J. Mashreghi, Admissible majorants for model subspaces of $ H^2$. I. Slow winding of the generating inner function, Canad. J. Math. 55 (2003), 1231-1263. MR 2016246 (2004i:30029a)
  • [HMII] -, Admissible majorants for model subspaces of $ H^2$. II. Fast winding of the generating inner function, Canad. J. Math. 55 (2003), 1264-1301. MR 2016247 (2004i:30029b)
  • [Hof] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1962. MR 0133008 (24:A2844)
  • [Koo1] P. Koosis, The logarithmic integral. I, Cambridge Stud. Adv. Math., vol. 12, Cambridge Univ. Press, Cambridge, 1988. MR 0961844 (90a:30097)
  • [Koo2] -, The logarithmic integral. II, Cambridge Stud. Adv. Math., vol. 21, Cambridge Univ. Press, Cambridge, 1992. MR 1195788 (94i:30027)
  • [Koo3] -, Leçons sur le théorème de Beurling et Malliavin, Univ. Montréal, Les Publications CRM, Montréal, QC, 1996. MR 1430571 (99e:42023)
  • [Koo4] -, Introduction to $ H^p$ spaces. With an appendix on Wolff's proof of the corona theorem, London Math. Soc. Lecture Note Ser., vol. 40, Cambridge Univ. Press, Cambridge-New York, 1980. MR 0565451 (81c:30062)
  • [Koo5] -, Harmonic estimation in certain slit regions and a theorem of Beurling and Malliavin, Acta Math. 142 (1979), 275-305. MR 0521462 (80d:31007)
  • [Koo6] -, La plus petite majorante surharmonique et son rapport avec l'existence des fonctions entières de type exponentiel jouant le rôle de multiplicateurs, Ann. Inst. Fourier (Grenoble) 33 (1983), 67-107. MR 0698850 (84k:30032)
  • [Koo7] -, A local estimate, involving the least superharmonic majorant, for entire functions of exponential type, Algebra i Analiz 10 (1998), no. 3, 45-64; English transl., St. Petersburg Math. J. 10 (1999), no. 3, 441-455. MR 1628022 (99g:30035)
  • [Koo8] -, A result on polynomials and its relation to another concerning entire functions of exponential type, Mat. Fiz. Anal. Geom. 5 (1998), no. 1/2, 68-86. (English) MR 1631826 (99d:30005)
  • [KooP] P. Koosis and H. L. Pedersen, Lower bounds on the values of an entire function of exponential type at certain integers, in terms of a least superharmonic majorant, Algebra i Analiz 10 (1998), no. 3, 31-44; English transl., St. Petersburg Math. J. 10 (1999), no. 3, 429-439. MR 1628018 (99g:30034)
  • [Kr] M. G. Krein, A contribution to the theory of entire functions of exponential type, Izv. Akad. Nauk SSSR Mat. 11 (1947), no. 4, 309-326. (Russian) MR 0022252 (9:179e)
  • [L] N. N. Luzin, On a certain integral, Integral and Trigonometric Series, GITTL, Moscow, 1951, pp. 287-319. (Russian) MR 0048364 (14:2g)
  • [Ls] N. Levinson, Gap and density theorems, Amer. Math. Soc. Colloq. Publ., vol. 26, Amer. Math. Soc., New York, 1940. MR 0003208 (2:180d)
  • [Lv] B. Ya. Levin, Distribution of zeros of entire functions, Gostekhizdat, Moscow, 1956; English transl., Transl. Math. Monogr., vol. 5, Amer. Math. Soc., Providence, RI, 1980. MR 0087740 (19:402 c); MR 0589888 (81k:30011)
  • [MGLP] B. M. Makarov, A. A. Lodkin, A. N. Podkorytov, and M. G. Goluzina, Selected problems in real analysis, ``Nevskii Dialekt,'' St. Petersburg, 2004. (Russian).
  • [MIT] G. G. Magaril-Il'yaev and V. M. Tikhomirov, On inequalities for derivatives of Kolmogorov type, Mat. Sb. 188 (1997), no. 12, 73-106; English transl., Sb. Math. 188 (1997), no. 12, 1799-1832. MR 1607438 (98m:41022)
  • [MP] N. Makarov and A. Poltoratski, Meromorphic inner functions, Toeplitz kernels, and the uncertainty principle (to appear).
  • [Mal] P. Malliavin, On the multiplier theorem for Fourier transforms of measures with compact support, Ark. Mat. 17 (1979), 69-81. MR 0543504 (81e:42023)
  • [N] N. Nikol'skii, Treatise on the shift operator. Spectral function theory, Grundlehren Math. Wiss., vol. 273, Springer-Verlag, Berlin, 1986. MR 0827223 (87i:47042)
  • [P] H. L. Pedersen, Entire functions and logarithmic sums over nonsymmetric sets of the real line, Ann. Acad. Sci. Fenn. Math. 25 (2000), 351-388. MR 1762422 (2001f:30034)
  • [Pr] I. I. Privalov, Boundary properties of analytic functions, 2nd ed., GITTL, Moscow-Leningrad, 1950. (Russian) MR 0047765 (13:926h)
  • [Rem] Ch. Remling, Schrödinger operators and de Branges spaces, J. Funct. Anal. 196 (2002), 323-394. MR 1943095 (2003j:47055)
  • [Ru] W. Rudin, Fourier analysis on groups, Intersci. Tracts Pure Appl. Math., No. 12, Intersci. Publ., New York-London, 1962. MR 0152834 (27:2808)
  • [T] V. M. Tikhomirov, Some questions in approximation theory, Moskov. Univ., Moscow, 1976. (Russian) MR 0487161 (58:6822)
  • [Ti] E. Titchmarsh, Introduction to the theory of Fourier integrals, Gostekhizdat, Moscow-Leningrad, 1948; English transl., Chelsea Publ. Co., New York, 1986. MR 0942661 (89c:42002)
  • [Wor] H. Woracek, de Branges spaces of entire functions closed under forming difference quotients, Integral Equations Operator Theory 37 (2000), 238-249. MR 1769812 (2001g:46058)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 42A50, 30D55

Retrieve articles in all journals with MSC (2000): 42A50, 30D55

Additional Information

J. Mashreghi
Affiliation: Département de Mathématiques et de Statistique, Université Laval, Laval, Québec G1K7P4, Canada

F. L. Nazarov
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48821

V. P. Havin
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospect 28, Staryĭ Peterhof, St. Petersburg 198904, Russia

Keywords: Fourier transform, spectrum, Hardy space, Paley--Wiener space, Hilbert transform, inner function, outer function, logarithmic integral, Beurling--Malliavin theorem.
Received by editor(s): March 20, 2005
Published electronically: July 20, 2006
Additional Notes: This work was supported by RFBR (grant no. 01-01-00377) and by “Scientific Schools” grant no. Sh-2266.2003.1
Dedicated: In fond memory of Ol$’$ga Aleksandrovna Ladyzhenskaya
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society