On an analogue of the Kotel’nikov-Shannon formula for homogeneous and isotropic random fields. (English. Russian original) Zbl 0632.60045

Omarov, S. O.; Yadrenko, M. I.

Theory Probab. Math. Stat. 35, 83-85 (1987); translation from Teor. Veroyatn. Mat. Stat. 35, 76-78 (1986).

The result of the paper is the proof of the formula \[ \xi (r,\phi)= \]
\[ (4/n)\sum^{\infty}_{k=1}\sum^{n-1}_{m=0}\sum^{n- 1}_{p=0}\frac{J_ 0(Rr)}{[\lambda^ 2_ k-(Rr)^ 2]J_ 1(\lambda_ k)}(\frac{Rr}{\lambda_ k})^ pe^{ip}(\phi -2\pi m/n)\xi (\lambda_ k/R,\phi_ m), \] where \(\phi_ m=2\pi m/n\) \((m=0,1,...,n- 1)\), \(\xi\) (r,\(\phi)\) is a homogeneous and isotropic random field on the plane, whose spectrum is interior to the disk of radius R, \(J_ 0(x)\) and \(J_ 1(x)\) are the Bessel functions and \(\{\lambda_ n\}\) is the sequence of roots of the equation \(J_ 0(x)=0\).

Reviewer: A.N.Radchenko

MSC:

60G60	Random fields
60G10	Stationary stochastic processes

Keywords:

homogeneous and isotropic random field on the plane; Bessel functions