2024/01/15 by Komla Domelevo, Polona Durcik, Domelevo, Komla +11
Computer Science · Mathematics · #41A63 #47A60 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #Numerical methods in inverse problems #Primary 42C10 #Secondary 41A17
paper · pdf · doi:10.48550/arxiv.2401.07699
openalex publication_date 2024/01/15 · openalex created_date 2024/01/19 · openalex updated_date 2026/08/01
The main result of this paper are dimension-free Lp inequalities, 12, ε>0, and θ=θ(ε,p)∈ (0,1) satisfying (1)/(p)=\fracθp+ε+(1-θ)/(2) we obtain, for any function f:\-1,1\n→ ℂ whose spectrum is bounded from above by d, the Bernstein-Markov type inequalities ‖Δk f‖p ≤ C(p,ε)k dk ‖f‖21-θ‖f‖p+εθ, k∈ ℕ. Analogous inequalities are also proved for p∈ (1,2) with p-ε replacing p+ε. As a corollary, if f is Boolean-valued or f\colon \-1,1\n→ \-1,0,1\, we obtain the bounds ‖Δk f‖p ≤ C(p)k dk ‖f‖p, k∈ ℕ. At the endpoint p=∞ we provide counterexamples for which a linear growth in d does not suffice when k=1. We also obtain a counterpart of this result on tail spaces. Namely, for p>2 we prove that any function f:\-1,1\n→ ℂ whose spectrum is bounded from below by d satisfies the upper bound on the decay of the heat semigroup ‖e-tΔf‖p ≤ exp(-c(p,ε) td) ‖f‖21-θ‖f‖p+εθ, tgt;0, and an analogous estimate for p∈ (1,2). The constants c(p,ε) and C(p,ε) depend only on p and ε; crucially, they are independent of the dimension n.