A survey on uncertainty principles related to quadratic forms

• Autores: Bruno Demange, Aline Bonami
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 57, Fasc. Extra 1, 2006 (Ejemplar dedicado a: Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations), págs. 1-36
• Idioma: español
• Enlaces
  • Texto completo
• Resumen

  • Hardy¿s Uncertainty Principle may be seen as a characterisation of all tempered distributions f on Rd such that e±qf and e±q0 bf 2 S0(Rd) are also tempered, with q and q0 two positive definite quadratic forms.

    After proving this, we consider the same problem for general non degenerate quadratic forms q and q0. A special attention is given to the case when q(x, y) = q0(x, y) = hx, yi on Rd × Rd, for which the results that we describe here may be seen as a generalization of Beurling¿s Uncertainty principle. We also consider other kinds of uncertainty principles related to quadratic forms, which describe strong or weak annihilating pairs in the sense of Havin and J¿oricke.

• Referencias bibliográficas
  • W.O. Amrein and A.M. Berthier, On support properties of $L^p$-functions and their Fourier transforms, J. Funct. Anal. 24 (1977), 258–267.
  • C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, and G. Scheffer, Sur les Inégalités de Sobolev Logarithmiques,...
  • G.F. Bachelis and S.E. Ebenstein, On $\Lambda(p)$ sets, Pacific J. Math. 54 (1974), 35–38.
  • V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187–214.
  • V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, II: A family of related function spaces, Application...
  • M. Benedicks, On Fourier transforms of functions supported on sets of finite Lebesgue measure, J. Math. Anal. Appl. 106 (1985), 180–183.
  • A. Bonami, B. Demange, and P. Jaming, Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms, Rev....
  • J. Bourgain, Bounded orthogonal systems and the $\Lambda(p)$-set problem, Acta Math. 162 3-4 (1989), 227–245.
  • M.G. Cowling and J.F. Price, Generalisations of Heisenberg’s Inequality, Springer, Berlin, 1983.
  • B. Demange, Uncertainty principles for the ambiguity function, Proc. London Math. Soc., to appear.
  • B. Demange, Uncertainty inequalities associated to some homogeneous functions, to appear in C. R. Acad Sciences Paris.
  • B. Demange, Uncertainty principles for quadratic forms (mémoire), to appear.
  • B. Demange, Principes d’incertitude associés à des formes quadratiques non dégénérées, thesis, Orléans, 2004.
  • G.B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, NJ, 1989.
  • G.B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (1997), 207–238.
  • I.M. Gel'fand and G.E. Shilov, Fourier transforms of rapidly increasing functions and questions of uniqueness of the solution of Cauchy's...
  • I.M. Gel'fand and G.E. Shilov, Generalized Functions 2, Moscow, 1958 (Russian), Academic Press, New York-London, 1968.
  • K. Gröchenig, Uncertainty principles for time-frequency representations, Appl. Numer. Harmon. Anal. (2003), 11–30.
  • K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhauser, Boston, 2000.
  • K. Gröchenig and G. Zimmermann, Hardy's theorem and the short-time Fourier transform of Schwartz functions, J. London Math. Soc. (2) 63...
  • G.H. Hardy, A theorem concerning Fourier transforms, J. London Math. Soc. 8 (1933), 227–231.
  • K.E. Hare, Strict-2-associatedness for thin sets, Colloq. Math. 56 (1988), 367–381.
  • K.E. Hare, Arithmetic properties of thin sets, Pacific J. Math. 131 (1988), 143–155.
  • V.P. Havin, On the uncertainty principle in harmonic analysis, Twentieth century harmonic analysis, a celebration, (Il Ciocco, 2000, Italy),...
  • V.P. Havin and B. Jöricke, The Uncertainty Principle in Harmonic Analysis, Springer-Verlag, Berlin, 1994.
  • L. Hörmander, A uniqueness theorem of Beurling for Fourier transform pairs, Ark. Mat. 29 (1991), 237–240.
  • O. Kovrijkine, Some results related to the Logvinenko-Sereda theorem, Proc. Amer. Math. Soc. 129 (2001), 3037–3047.
  • O. Kovrizhkin, A version of the uncertainty principle for functions with lacunary Fourier transforms, J. Math. Anal. Appl. 288 (2003), 606–633.
  • B.J. Levin, Distribution of Zeros of Entire Functions, Translations of Mathematical Monographs 5, A.M.S., Providence, R.I., 1980.
  • I.M. Miheev, Trigonometric series with gaps, Anal. Math. 9 (1983), 43–55.
  • I.M. Miheev, Series with gaps, (Russian) Mat. Sb. (N.S.) 98(140) (1975), 538–563, 639.
  • G.W. Morgan, A note on Fourier transforms, J. London Math. Soc. 9 (1934), 187–192.
  • F.L. Nazarov, Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type, Algebra...
  • R.P. Sarkar and J. Sengupta, Beurling’s theorem and characterization of heat kernel for Riemannian symmetric spaces of noncompact type authors,...
  • C. Shubin, R. Vakilian, and T. Wolff, Some harmonic analysis questions suggested by Anderson-Bernoulli models, Geom. Funct. Anal. 8 (1998),...
  • A. Zygmund, On a theorem of Hadamard, Ann. Soc. Polon. Math. 21 (1948), 52–69.