1. Typically in I. J. Schoenberg’s theory of variation-diminishing transforms; ef. Widder, D. V., The Convolution Transform(Princeton: Princeton Univ. Press, 1955 ). Note that the smoothing operators on the real line considered above are indeed variation-diminishing on continuous functions.
2. After Reynolds, Osborne; cf. Phil. Trans. Roy. Soc., A186, part I, 123–164 (1895). For a survey of the work up to 1957, cf. Iubreil-Jacotin, M.-L., Etude Algébrique des transfo r mations de Reynolds, Colloque d’Algèbre Supérieure( Louvain: CBRS, 1957 ), pp. 9 - 27.
3. Cf. Kampé, J., de Fériet, “Fonctions aléatories et théorie statistique de la turbulence,” in Theorie des Fonctions Aléatoires, ed. A. Blanc-Lapierre et R. Fortet, (Paris: Masson, 1053). Cf. also 1)ubreil-Jacotin, M.-L., “Sur le Passage des Equations de Navier-Stokes aux Equations de Reynolds,” C. R. Acad. Sci., Paria, 244, 2887–2890(1957).
4. Cf. Loeve, Probability Theory (New York: Van Nostrand, 1955), P. 341 ff.
5. Cf. Arbault, J., “Transformations de Reynolds Bur un ensemble fini,” Corso sulla teoria delia turbolenza(Turin, 1957), pp. 115–121; Billik, M., and G.-C. Rota, “On Reynolds Operators in Finite-Dimensional Algebras,” J. Math. and Mechanics, 9, (1960);