thc rclittic~ti hctwccn thc Lagranyian ant1 thc Eulerian power spectra. Replitcing /' , I I I 1 . ~ 1 . (10) by its cosinc tr;insl.orm o w obtains the [-agrangian
powcr upcctrirni cxprcssed by the Kulcriiin correlation function
I
, t f ( r ) = \ .s-- I T(.C I ) cos(r,s) cis
* 0
(12)
a iid
I
M(0) = [ s ' 7 ( s ' I ) (1s = /1,,'/,:
the ratio of the Lagrangkin and the Euleriim integral scale is easy to deduce
from Eqs. (10) and (12).
The transfer functions L and A4 are connected by the self-reciprocal divisor transform (see Oberhettinger, 1972, for pairs of transformation)
' 0
(13)
tvith rhc modificd and the ordinary Hessel function of the second kind and
zero order. Of course cos(s) is one of the cigenfunctions of Eq. (14).
Starting from the inversion of Eq. (3) with the transfer function T* as the
resolvent of T . Eqs. (6)--( 14) are all followed with L* as the resolvent of M
and M* the resolvent of L.
The relations between the transfer functions T. L, M , and their resolvents
Lire simple only in implicit forms:
" 0
= A(r - 1)
with thc Dirac delta function, and
I
1
T*(s)L(rs) ds = I .\-- I T*(s)M(r.s- ') ds
. c,
* 0
(16)
= cos( r).
I3y the traiislcr hmction 7' there are further given the relations between the
exclisrigc coclficicnt A and thc variancc 6 ' (both deiined in the Lagrangian
system) on the one side and the corrcsponding functions derived formally