Empirical approximation of Lᵖ norms for p∈ (1, 2)

For p∈[1,2], we prove uniform moment estimates for weighted sums \[ ∑ⱼ₌₁^mζ_jw_j|{ v}_j,y|^p \] of independent subgaussian random variables ζⱼ, where the unit vectors \ vⱼ\ⱼ₌₁ᵐ and nonnegative weights ⱼ\ⱼ₌₁ᵐ form an isotropic frame. This allows us to obtain optimal sampling discretization of Lₚ norms for each fixed p∈[1,2]. Namely, for an N-dimensional space XN of bounded functions satisfying \|f\|_∞≤√KN\,\|f\|L₂ for every f∈ XN, we show that m≥ Cε⁻²KNlog(2N) independent samples ξ¹,…,ξᵐ provide the discretization \[ (1-ε)\|f\|L_p^p ≤1m∑ⱼ₌₁^m|f(ξ^j)|^p ≤(1+ε)\|f\|L_p^p ∀ f∈ X_N \] with probability at least 1-exp(-cmε²/KN). This implies a corresponding weighted discretization result that, for $1