Distribution of particles which produces a desired radiation pattern

If Aq(β,α,k) is the scattering amplitude, corresponding to a potential qL2(D), where DR3 is a bounded domain, and eikα.x is the incident plane wave, then we call the radiation pattern the function A(β)Aq(β,α,k), where the unit vector α, the incident direction, is fixed, and k>0, the wavenumber, is fixed. It is shown that any function f(β)L2(S2), where S2 is the unit sphere in R3 and β is the scattering direction, can be approximated with any desired accuracy by a radiation pattern: f(β)-A(β)L2(S2)<ε, where ε>0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and ε. There is a one-to-one correspondence between the above potential and the density of the number of small acoustically soft particles DmD, 1=0, arbitrarily close in the norm of L2(S2xS2) to an arbitrary given scattering amplitude f(α',α), corresponding to a real-valued potential qL2(D), i.e., corresponding to an arbitrary given refraction coefficient in D