Stationary functions and their applications to the theory of turbulence

Abstract The Monte-Carlo method makes use of sequences of “random numbers” which are often defined by purely arithmetical properties. That suggests the interpretation of the Monte-Carlo method in analytical, nonstatistical terms. Random numbers are replaced by “uniformly distributed sequences”. A short study of such sequences and their realizations is made. Similarly, it happens that random functions can be replaced by nonrandom functions, defined by properties of “temporal mean values,” like Mf = lim T→∞ 1 2T ∫ −T −T f(t) dt . Functions for which Mf and M f (t) f(t + τ) (the correlation function of f) exist are called stationary functions. A study is made of the correlation function γ(τ), which is a positive-definite function, the Fourier transform of a positive bounded measure. The class of stationary functions contains, essentially, almost periodic functions and pseudorandom functions, for which limτ → ∞ γ(τ) = 0. Processes are given for the construction of pseudorandom functions, involving uniformly distributed sequences. Through convolution by an integrable function a stationary function f is transformed into a stationary function of the same category, but possibly more regular (continuous, differentiable,…) than f. The class of all stationary functions does not have a good algebraic structure, but can be embedded in a Banach space, the Marcinkiewicz space, and contains linear subspaces. The most important of these subspaces is generated by the translation of a given stationary function; in this space a harmonic analysis is possible. Some final remarks are made about the “asymptotic measure,” i.e., the distribution of the values of a stationary function, and the effect of a change of scale. In this paper, only some elementary proofs are given. In the appendices, the reader will find the proofs of those essential theorems which are not contained in the main text. Nevertheless the proofs of a number of useful theorems are too long and too technical to be developed here. References are given, in which the reader will find all explanations he may wish. This paper will be followed by a second one, in which the theory of stationary functions will be applied to the construction of turbulent solutions of Navier-Stokes equations.