To generalize Theorem 5.11.1 to other designs, let Z =

(Z1,...,ZN ) and let G = {g1,...,gr} be a group of permutations of N coordinates or more generally a group of orthogonal transformations of N-space If Pσ,ζ(z) = 1 r

r k=1 1

(

√2πσ)N exp

− 1 2σ2 |z − gkζ|

2



, (5.80)

where |z|

2 = z2 i , then φ(z)pσ,ζ(z) dz ≤ α for all σ > 0 and all ζ implies 1

r



z
∈S(z)

φ(z

) ≤ α a.e., (5.81)

where S(z) is the set of points in N-space obtained from z by applying to it all the transformations gk, k = 1,...,r.