Kraichnan 

 The coefficients of the power- series expansion of f(\) are then the moments 



r 



I p (a) a" da . 



The Pade table approximates the unknown p{a) by a weighted sum of S-f unctions, 

 S. p. (a - a. ), such that, as we go higher in the table, more and more moments 

 are reproduced correctly [36]. Expansion of a function in S-functions is a spe- 

 cial case of expansion in a complete set of orthogonal functions, and Eq. (24) is 

 a special case of an integral representation of the form 



f(\) = J P (a) g (a\2) da , 

 



where g(a^^) has a power-series expansion. Thus a number of generalizations 

 of the Pade approximants are possible. 



If p in Eq. (24) is nonnegative, a case which includes Eq. (22), then the ap- 

 proximants on the diagonal of the Pade table yield successively improving upper 

 bounds on f(A.), while a set of approximants off the diagonal yield successively 

 improving lower bounds. It should be noted that the approximation to f(x), ob- 

 tained by substituting a sum of s-functions for p{a), is a smooth function. 



The technique of Pade approximation, and its generalizations, can be applied 

 formally to the infinite expansions of turbulence theory by regarding the latter 

 as a power series in a parameter. In the case of perturbation expansion of tur- 

 bulence functions about purely viscous decay values, the ordering parameter is 

 the Reynolds number. In the case of Eq. (3), we can form the power series by 

 multiplying the successively higher terms on the right-hand side by powers of ^^^ 

 and, at the end, taking \ = 1. Here \ is a formal ordering parameter without 

 immediate significance. 



To justify such manipulations, we must establish that appropriate forms of 

 integral representation exist for the functions of interest. Only a start at this 

 has been made at the time of writing. The plausibility of representations like 

 Eq. (24) perhaps is enhanced by the nature of turbulence functions as averages 

 over an ensemble of realizations, with p playing the role of probability distribu- 

 tion for an actual or effective parameter. Then large values of "a" would cor- 

 respond to contributions from the fringes of the probability distribution, which 

 have little effect on the final values of the functions of interest but which affect 

 strongly the higher terms in the divergent power series. 



The Pade technique, and some generalizations, has been tried out with ap- 

 parent success on several turbulence problems, including a detailed application to 

 Eq. (10) [37j. The theory of successive approximations for the Laplace transform 

 of G(k, t) has been worked out fully in the limit of high k, and the Pade approxi- 

 mants have been shown to yield bounds on errors, as well as improving approxi- 

 mations in this limit. The theory is in less complete shape for general k, but 

 comparison with the computer experiments show that improvement over the 

 direct-interaction results is obtained at all k, with a reduction of about 50% in 



808 



