Proposition

Theorem 4.1 ([Uniform representing densities]). There is a constant K_*>0 , depending only on the fixed profiles f,b and R_0 , with the following property. For every fixed D_0>1,2√2R_0 there is an integer threshold Q_0 such that, for every …

Theorem 4.1 ([Uniform representing densities]). There is a constant K_*>0 , depending only on the fixed profiles f,b and R₀ , with the following property. For every fixed D₀>1,2√2R₀ there is an integer threshold Q₀ such that, for every integer Q≥Q₀ and every choice of the separated sets above, the recursion has the following properties for all j≥1 : W_j is smooth and strictly positive, and W_j\,dσ represents holomorphic evaluation at zero. In particular it is a probability measure. Under global phase rotation, W_j has frequencies only between -ℓ_j and ℓ_j , and its mean on every phase orbit is exactly one. At every point of S , c_{0W_} j-1 ≤W_{j≤2W_} j-1 ,; |ZW_j|≤K_*√k_j\,W_j, |TW_j|≤K_*k_jWj for every unit horizontal vector Z . The threshold Q₀ may also be enlarged to impose any fixed lower threshold on the degrees k_j .