Geodesic
Unbounded Positive Definite Functions
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1309-1318

Voir la notice de l'article provenant de la source Cambridge University Press

Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality 1 holds for every choice of complex numbers C1, ..., cn and S1, ..., sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p. 122) and Raïkov (5) proved that every continuous positive definite function on a locally compact abelian group is the Fourier-Stieltjes transform of a bounded positive measure, thus generalizing theorems of Herglotz (4) (G = Z, the integers) and Bochner (1) (G = R, the real numbers).If ƒ is a continuous function, then condition (1) is equivalent to the condition that 2 
Stewart, James. Unbounded Positive Definite Functions. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1309-1318. doi: 10.4153/CJM-1969-143-4
@article{10_4153_CJM_1969_143_4,
     author = {Stewart, James},
     title = {Unbounded {Positive} {Definite} {Functions}},
     journal = {Canadian journal of mathematics},
     pages = {1309--1318},
     year = {1969},
     volume = {21},
     number = {1},
     doi = {10.4153/CJM-1969-143-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-143-4/}
}
TY  - JOUR
AU  - Stewart, James
TI  - Unbounded Positive Definite Functions
JO  - Canadian journal of mathematics
PY  - 1969
SP  - 1309
EP  - 1318
VL  - 21
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-143-4/
DO  - 10.4153/CJM-1969-143-4
ID  - 10_4153_CJM_1969_143_4
ER  - 
%0 Journal Article
%A Stewart, James
%T Unbounded Positive Definite Functions
%J Canadian journal of mathematics
%D 1969
%P 1309-1318
%V 21
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1969-143-4/
%R 10.4153/CJM-1969-143-4
%F 10_4153_CJM_1969_143_4

[1] 1. Bochner, S., Vorlesungen uber Fourier sche Integrate (Akademische Verlagsgesellschaft, Leipzig, 1932) ; translated by Morris Tenenbaun and Harry Pollard: Lectures on Fourier integrals (with the author's supplement on Monotonie functions, Stieltjes integrals, and harmonic analysis), Annals of Mathematics Studies, No. 42 (Princeton Univ. Press, Princeton, N.J., 1959). Google Scholar

[2] 2. Cooper, J. L. B., Positive definite functions of a real variable, Proc. London Math. Soc. (3) 10 (1960), 53–66. Google Scholar

[3] 3. Gel'fand, I. M. and Vilenkin, N. Ya., Generalized functions, Vol. 4, Applications of harmonic analysis (Academic Press, New York, 1964). Google Scholar

[4] 4. Herglotz, G., Über Potenzreihen mit positiven reelen Teil im Einheitskreis, Leipziger Berichte 63 (1911), 501–511. Google Scholar

[5] 5. Raïkov, D. A., Positive definite functions on commutative groups with an invariant measure, C. R. (Doklady) Acad. Sci. URSS (N. S.) 28 (1940), 296–300. (Russian) Google Scholar

[6] 6. Rudin, W., Fourier analysis on groups, Interscience Tracts in Pure and Applied Math., No. 12 (Interscience, New York, 1962). Google Scholar

[7] 7. Simon, A. B., Cesàro summability on groups: Characterization and inversion of Fourier transforms, pp. 208-215 in Function algebras, Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965 (Scott-Foresman, Chicago, Illinois, 1966). Google Scholar

[8] 8. Stewart, J. D., Positive definite functions and generalizations, Ph.D. thesis, University of Toronto, Toronto, Ontario, 1967. Google Scholar

[9] 9. Weil, André, L'intégration dans les groupes topologiques et ses applications, Actualités Sci. Indust., no. 869 (Hermann, Paris, 1940; revised, 1951). Google Scholar

Cité par Sources :