×

zbMATH — the first resource for mathematics

Superadditivity of Fisher’s information and logarithmic Sobolev inequalities. (English) Zbl 0732.60020
Summary: We prove a theorem characterizing Gaussian functions and we prove a strict superadditivity property of the Fisher information. We use these results to determine the cases of equality in the logarithmic Sobolev inequality on \(R^ n\) equipped with Lebesgue measure and with Gauss measure. We also prove a strengthened form of Gross’s logarithmic Sobolev inequality with a “remainder term” added to the left side. Finally we show that the strict form of Gross’s inequality is a direct consequence of an inequality due to Blachman and Stam, and that this in turn is a direct consequence of strict superadditivity of the Fisher information.

MSC:
60E15 Inequalities; stochastic orderings
94A15 Information theory (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bakry, D; Emery, M, Diffusions hypercontractives, (), 179-206 · Zbl 0561.60080
[2] Barron, A.R, Entropy and the central limit theorem, Ann. probab., 14, 336-342, (1986) · Zbl 0599.60024
[3] Beckner, W, Inequalities in Fourier analysis, Ann. of math., 102, 159-182, (1975) · Zbl 0338.42017
[4] Bernstein, S, Sur une caractéristique de la loi de Gauss, Trans. leningrad polytech. inst., 3, 21-22, (1941)
[5] Blachman, N.M, The convolution inequality for entropy powers, IEEE trans. inform. theory, 2, 267-271, (1965) · Zbl 0134.37401
[6] Brascamp, H.J; Lieb, E.H, Best constant in Young’s inequality, its converse, and its generalization to more than three functions, Adv. in math., 20, 151-173, (1976) · Zbl 0339.26020
[7] Brothers, J.E; Zeimer, W.P, Rearrangement and minimal Sobolev functions, J. reine angew. math., 348, 153-179, (1988)
[8] Carlen, E.A, Some integral identities and inequalities for entire functions and their application to the coherent state transform, (1989), Princeton preprint
[9] {\scE. A. Carlen and M. Loss}, Extremals of functionals with competing symmetries, J. Funct. Anal., in press. · Zbl 0705.46016
[10] Cramér, H, Über eine eigenschaft der normalen verteilungsfunktion, Math. Z., 41, 405-414, (1936) · JFM 62.0597.02
[11] fisher, R.A, Theory of statistical estimation, (), 700-725 · JFM 51.0385.01
[12] Gross, L, Logarithmic Sobolev inequalities, Amer. J. math., 97, 1061-1083, (1975) · Zbl 0318.46049
[13] Hardy, G.H, A theorem concerning Fourier transforms, J. London math. soc., 8, 227-231, (1933) · Zbl 0007.30403
[14] Hirschman, I.I, A note on entropy, Amer. J. math., 79, 152-156, (1957) · Zbl 0079.35104
[15] Bialnicki-Birula, I; Mycielski, J, Entropy and the uncertainty principle, Comm. math. phys., 44, 129-136, (1975)
[16] Bialnicki-Birula, I; Mycielski, J, Nonlinear wave mechanics, Ann. physics, 100, 62-97, (1976)
[17] Kac, M, On a characterization of the normal distribution, Amer. J. math., 61, 473-476, (1939) · Zbl 0020.37603
[18] Lévy, P, Processus stochastique et mouvement brownien, (1948), Gauthier-Villars Paris
[19] Lieb, E.H, Proof of an entropy conjecture of wehrl, Comm. math. phys., 62, 35-41, (1978) · Zbl 0385.60089
[20] Lieb, E.H, Gaussian kernels have Gaussian maximisers, (1989), Princeton preprint
[21] Segal, I.E, Tensor algebras over Hilbert spaces, I, Trans. amer. math. soc., 81, 106-134, (1956) · Zbl 0070.34003
[22] Skitovich, V.P, Linear forms of independent random variables and the normal distribution, Izv. akad. nauk USSR, 18, 185-200, (1954)
[23] Stam, A.J, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Inform. and control, 2, 101-112, (1959) · Zbl 0085.34701
[24] Weissler, F.B, Logarithmic Sobolev inequalities for the heat-diffusion semigroup, Trans. amer. math. soc., 237, 255-269, (1978) · Zbl 0376.47019
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.