Fonctions opérant sur les fonctions définies-positives
Annales de l'Institut Fourier, Tome 13 (1963) no. 1, pp. 161-180.

Soit G un groupe commutatif localement compact. On se propose de déterminer les fonctions f, définies sur le disque-unité ouvert du plan complexe [z:|z|<1], à valeurs complexes, telles que la fonction composée f(ϕ) soit définie-positive chaque fois que ϕ est une fonction définie-positive sur G avec |ϕ|<1 partout. On prouve que si G contient des éléments dont les ordres sont aussi grands qu’on veut, alors il faut et il suffit que f soit représentée par une série convergente pour |z|<1

f(x)=∑m,n=0∞Am,nZmZ¯noùchaqueAm,n≥0.

Puis on étudie quelques problèmes voisins mais plus fins que celui-ci.

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Herz, Carl S. Fonctions opérant sur les fonctions définies-positives. Annales de l'Institut Fourier, Tome 13 (1963) no. 1, pp. 161-180. doi : 10.5802/aif.137. https://www.numdam.org/articles/10.5802/aif.137/

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[4] J. Schur, Bemerkungen zur Theorie des beschränkten Bilineanformen mit unendlichen vielen Veranderlichen, J. für. Math., 140, 1911, p. 14. | JFM

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