Cybenko

記号
 denotes the n-dimensional unit cube
 the space of continuous functions on In
 the supremum norm of an f∈C(In)

Th. 1
Let σ be any continuous discriminatory function. Then finite sums of the form

are dense in C(In). In other words, given any f∈C(In) and ε>0, there is a sum, G(x),
of the above form, for which

Th. 2
Let σ be any continuous sigmoidal function. Then finite sums of the form

are dense in C(In). In other words, given any f∈C(In) and ε>0, there is a sum, G(x),
of the above form, for which

Th. 4
Let σ be bounded measurable sigmoidal function. Then finite sums of the form

are dense in L1(In). In other words, given any f∈L1(In) and ε>0, there is a sum, G(x),
of the above form, for which