Optimum quantization and its applications

Advances in Mathematics

Volumn 186, Issue 2, 2004, Pages 456-497

  Gruber, Peter M ^a  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

Author keywords

Approximation of probability measures; Best approximating polytopes; Distortion of vector quantizers; Error of numerical integration formulae; Minimum isoperimetric quotient; Optimum quantization; Sums of moments

Indexed keywords

EID: 3042857517     PISSN: 00018708     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.aim.2003.07.017     Document Type: Article

References (48)

1. Babenko V.F. Asymptotically sharp bounds for the remainder for the best quadrature formulas for several classes of functions Mat. Zametki 19 1976 313-322; Math. Notes 19 (1976) 187-193
2. Babenko V.F. On the optimal error bound for cubature formulas on certain classes of continuous functions Anal. Math. 3 1977 3-9
3. Benson R.V. Euclidean Geometry and Convexity 1966 McGraw-Hill New York
4. p metric Israel J. Math. 117 2000 1-28
5. Brauner H. Differentialgeometrie 1981 Vieweg Braunschweig
6. Busemann H. The foundations of Minkowskian geometry Comment. Math. Helv. 24 1950 156-187
7. Chernaya E.V. An asymptotic sharp estimate for the remainder of weighted cubature formulas that are optimal on certain classes of continuous functions Ukraïn. Mat. Zh. 47 1995 1405-1415; Ukrainian J. Math. 47 (1995) 1606-1618
8. Chernaya (=Chornaya) E.V. On the optimization of weighted cubature formulae on certain classes of continuous functions East J. Approx. 1 1995 47-60
9. Conway J.H. Sloane N.J.A. Sphere packings, Lattices and Groups 3rd Edition 1999 Springer New York
10. Diskant V.I. Making precise the isoperimetric inequality and stability theorems in the theory of convex bodies Trudy Inst. Mat. (Novosibirsk) 14 1989 98-132
11. Diskant V.I. Generalizations of a theorem of Lindelöf Mat. Fiz. Anal. Geom. 4 1997 59-64
12. Du Q. Faber V. Gunzburger M. Centroidal Voronoi tessellations: applications and algorithms SIAM Rev. 41 1999 637-676
13. Encyclopedia of Mathematics, Kluwer Academic Publishers, Dordrecht, 1988-1994.
14. Ewald G. Larman D. Rogers C.A. The direction of the line segments and of the r-dimensional balls on the boundary of a convex body in Euclidean space Mathematika 17 1970 1-20
15. Fejes Tóth G. A stability criterion to the moment theorem Studia Sci. Math. Hungar. 34 2001 209-224
16. Fejes Tóth L. Sur la représentation d'une population infinie par un nombre fini d'elements Acta Math. Acad. Sci. Hungar. 10 1959 299-304
17. L. Fejes Tóth, Lagerungen in der Ebene, auf der Kugel und im Raum, 1st Edition, 1953, 2nd edition. Springer, Berlin, 1972.
18. Gersho A. Asymptotically optimal block quantization IEEE Trans. Inform. Theory IT-25 1979 373-380
19. Glasauer S. Gruber P.M. Asymptotic estimates for best and stepwise approximation of convex bodies III Forum Math. 9 1997 383-404
20. Graf S. Luschgy H. Foundations of quantization of Probability Distributions Lecture Notes Mathematics Vol. 1730 2000 Springer Berlin
21. Gray R.M. Neuhoff D.L. Quantization IEEE Trans. Inform. Theory IT-44 1998 2325-2383
22. Gray R.M. Neuhoff D.L. Omura J.K. Process definitions of distortion rate functions and source coding theorems IEEE Trans. Inform. Theory IT-21 1975 524-532
23. Gray R.M. Neuhoff D.L. Shields P.C. A generalization of Ornstein's d-bar distance with applications to information theory Ann. Probab. 3 1975 315-328
24. Gruber P.M. Volume approximation of convex bodies by circumscribed convex polytopes Discrete Math. Theor. Comput. Sci. 4 1991 309-317
25. P.M. Gruber, Aspects of approximation of convex bodies, in: P.M. Gruber, J.M. Wills (Eds.), Handbook of Convex Geometry A, North-Holland, Amsterdam, 1993, pp. 319-345.
26. Gruber P.M. Asymptotic estimates for best and stepwise approximation of convex bodies II Forum Math. 5 1993 383-404
27. Gruber P.M. Comparisons of best and random approximation of convex bodies by convex polytopes Rend. Sem. Mat. Univ. Palermo, Ser. II 50 Suppl. 1997 189-216
28. Gruber P.M. Optimal arrangements of finite point sets in Riemannian 2-manifolds Trudy Mat. Inst. Steklov. 225 1999 148-155; Proc. Steklov Math. Inst. 225 (1999) 160-167
29. Gruber P.M. A short analytic proof of Fejes Tóth's theorem on sums of moments Aequationes Math. 58 1999 291-295
30. Gruber P.M. Optimal configurations of finite point sets in Riemannian 2-manifolds Geom. Dedicata 84 2001 271-320
31. d Monatsh. Math. 135 2002 279-304
32. 3 Trudy Mat. Inst. Steklov. 239 2002 106-117; Proc. Steklov Math. Inst. 239 (2002) 96-107
33. Gruber P.M. Optimale quantisierung Math. Semesterber 49 2003 227-251
34. Hlawka E. The Theory of Uniform Distribution 1984 A B Academic Publishers Berkhamsted
35. Holmes R.D. Thompson A.C. N-dimensional area and content in Minkowski spaces Pacific J. Math. 85 1979 77-110
36. Kobayashi S. Nomizu K. Foundations of Differential Geometry 1969 Wiley New York
37. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities (Cauchy's equation and Jensen's inequality), PWN, Warszawa, 1985.
38. Lindelöf L. Propriétés générales des polyèdres qui, sous une étendue superficielle donnée, renferment le plus grand volume St. Petersburg Bull. Acad. Sci. 14 1869 258-269; Math. Ann. 2 (1870) 150-159
39. Matérn B. Persson O. On the extremum properties of the equilateral triangular lattice and the regular hexagonal network Förtech. Över. Rapp. Uppsala Stockholm 7 1965 15
40. Minkowski H. Allgemeine Lehrsätze über die convexen Polyeder Nachr. Kgl. Ges. Wiss. Göttingen Math.-Phys. Kl. 1897 1911 198-219; Ges. Abh. 2 (1911) 103-121
41. Neuhoff D.L. Gray R.M. Davisson L.D. Fixed rate universal block source coding with a fidelity criterion IEEE Trans. Inform. Theory IT-21 1975 511-523
42. Newman D.J. The hexagon theorem IEEE Trans. Inform. Theory IT-28 1982 137-139
43. Polovinkin V.I. Asymptotically optimal cubature weight formulas Sibirskii Mat. Zh. 70 1989 151 160; Siber. Math. J. 70 (1989) 289-297
44. Schneider R. Convex Bodies: The Brunn-Minkowski Theory 1993 Cambridge University Press Cambridge
45. Sobol' I.M. Quadrature formulas for functions in several variables that satisfy the general Lipschitz condition Zh. Vychisl. Mat. i Mat. Fiz. 29 1989 935-941; U.S.S.R. Comput. Math. Math. Phys. 29 (1989) 201-206
46. Sobolev S.L. Introduction to the Theory of Cubature Formulas 1974 Nauka Moscow
47. Thompson A.C. Minkowski Geometry 1996 Cambridge University Press Cambridge
48. Zador P.L. Asymptotic quantization error of continuous signals and the quantization dimension IEEE Trans. Inform. Theory IT-28 1982 139-148