Strict definiteness of integrals via complete monotonicity of derivatives

Transactions of the American Mathematical Society

Volumn 349, Issue 8, 1997, Pages 3321-3342

  Mattner, L ^a  

^a UNIVERSITY OF HAMBURG   (Germany)

Author keywords

Bernstein functions; Besicovitch covering theorem; Bilinear integrability; Conditionally positive definite functions; Determinate moment problem; Energy integrals; Integral inequalities; Logarithmic potential theory; Moment inequalities; Radial analysis

Indexed keywords

EID: 21744459346     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-97-01966-1     Document Type: Article

References (29)

1. R. Alexander and K.B. Stolarsky, Extremal problems of distance geometry related to energy integrals. Trans. Amer. Math. Soc. 193 (1974), 1-31. MR 50:3121
2. C. Berg, J. P. R. Christensen and P. Kessel, Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions. Springer, New York, 1984. MR 86b:43001
3. A. S. Besicovitch, A general form of the covering principle and relative differentiation of addititive functions. Proc. Cambridge Philos. Soc. 41 (1945), 103-110. MR 7:10e
4. R. N. Bhattacharya and R. Range Rao, Normal Approximation and Asymptotic Expansions. Wiley, New York, 1986. MR 8Tk:60062; MR 55:9219
5. G. Björk, Distributions of positive masses which maximize a certain generalized energy integral. Ark. Mat. 3 (1956), 255-269. MR 17:1198b
6. A. Buja, B. F. Logan, J. A. Reeds and L. A. Shepp, Inequalities and positive-definite functions arising from a problem in multidimensional scaling. Ann. Statist. 22 (1994), 406-438. MR 95a:60021
7. J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart. Springer, New York, 1984. MR 85k:31001
8. C. G. Esseen, Bounds for the absolute third moment. Scand. J. Statist. 2 (1975), 149-152. MR 52:4376
9. W. Feller, An Introduction to Probability Theory and Its Applications. Vol. 2, Second Edition. Wiley, New York, 1971. MR 42:5292
10. O. Frostmann, Potentiel d'équilibre et capacité des ensemble avec quelques applications a la théorie des fonctions. Thèse, Lund. Meddelanden Lunds Univ. Mat. Sem. 3 1935.
11. K. Guo, S. Hu and X. Sun, Conditionally positive definite functions and Laplace-Stieltjes integrals. J. Approx. Theory 74 (1993), 249-265. MR 94k:26015
12. E. Hille, Analytic Function Theory. Vol. 2. Blaisdell Publishing Company, Waltham, Massachusetts, 1962. MR 34:1490
13. _, Methods in Classical and Functional Analysis. Addison-Wesley, Reading, Massachusetts, 1972. MR 57:3802
14. W. C. M. Kallenberg, Problem 319. Statistica Neerlandica 49 (1995), 132.
15. N. S. Landkof, Foundations of Modern Potential Theory. Springer, Berlin, 1972. MR 50:2520
16. L. Mattner, Behandlung einiger Extrcmalprobleme für Wahrscheinlichkeitsverteilungen. Dissertation, Universität Hannover, 1990.
17. _, Completeness of location families, translated moments, and uniqueness of charges. Probab. Theory Related Fields 92 (1992), 137-149. MR 93f:60024
18. _, Extremal problems for probability distributions: a general method and some examples. In: Stochastic Inequalities, Moshe Shaked and Y. L. Tong, eds., IMS Lecture NotesMonograph Series Vol. 22, Inst. Math. Statist, Hayward, CA, 1992, pp. 224-283. MR 94c:60028
19. _, Bernstein's Theorem, inversion formula of Post and Widder, and the uniqueness theorem for Laplace transforms. Expo. Math.'ll (1993), 137-140. MR 94b:44001
20. C. A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2 (1986), 11-22. MR 88d:65016
21. G. Polya and G. Szegö, Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen, J. Reine Angew. Math. 165 (1931), 4-49.
22. Reprinted and commented on in: G. Polya (1974), Collected Papers, Vol. 1, Singularities of Analytic Functions, M.I.T. Press, Cambridge, Mass.,
23. and in: Gabor Szegö (1982), Collected Papers, Vol. 2., Birkhäuser, Boston. MR 58:21341; MR 84d:01082b
24. M. Riesz, Intégrales de Riemann-Liouville et potentiels. Ada Szeged Sect. Math. 9 (1937), 1-42.
25. Reprinted in: Marcel Riesz (1988), Collected Papers, Springer, Berlin, pp. 482-523. MR 90a:01109
26. \V. Rudin, Representation of functions by convolutions. J. Math, and Mech. 7 (1959), 103116. MR 21:3728
27. _, Real and Complex Analysis, Second Edition. McGraw-Hill, New York (1974). MR 49:8783
28. Z. Sasväri, Positive Definite and Definitizable Functions. Akademie Verlag, Berlin, 1994. MR 95c:43005
29. I. J. Schoenberg, Metric spaces and completely monotone functions. Ann. Math. 39 (1938), 811-841. Reprinted in: I. J. Schoenberg (1988), Selected Papers. Vol. 1. Birkhäuser, Boston, pp. 115-145. MR 91c:01051a