Positive Definite Functions in Quantum Mechanics and in Turbulence

Positive-definite functions have various representations which can be interpreted in probability theory, in quantam mechanics,and in turbulence theory. The equivalence of these representations is studied. The nature of the characteristic functions in quantum mechanics is discussed. In turbulence, time-correlations are positive-definite functions, but space-correlations have an ambiguous structure. In the annex, the properties of pseudo-random functions, which are convenient representations of turbulence, are summarized.