On the complexity of binary samples

Consider a class H of binary functions h: X → {-1,+1} on an interval X = [0, B] ⊂ IR. Define the sample width of h on a finite subset (a sample) S ⊂ X as ω_S (h) = min _x∈S |ω_h (x)| where ω_h (x)=h(x) max {a ≥ 0: h(z) = h(x), x - a ≤ z ≤ x + a}. Let double-struck S sign_ℓ be the space of all samples in X of ℓ and consider sets of wide samples, i.e., hypersets which are defined as A_β, h} = {S ∈ double-struck S sign_ℓ: ω_S(h) ≥ β}. Through an application of the Sauer-Shelah result on the density of sets an upper estimate is obtained on the growth function (or trace) of the class {A_β, h: h ∈ ℋ}, β > 0, i.e., on the number of possible dichotomies obtained by intersecting all hypersets with a fixed collection of samples S ∈ double-struk S sign_ℓ of cardinality m. The estimate is 2∑_i=0^{2⌊ B/(2β)⌋ (m-ℓ i).}