2024/02/02 by Gil Goldman, Goldman, Gil, Yosef Yomdin +1
Mathematics · #Advanced Harmonic Analysis Research #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.2402.01388
openalex publication_date 2024/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f: Bn → \mathbb R be a d+1 times continuously differentiable function on the unit ball Bn, with maxz∈ Bn |f(z)|=1. A well-known fact is that if f vanishes on a set Z⊂ Bn with a non-empty interior, then for each k=1,…,d+1 the norm of the k-th derivative ‖f(k)‖ is at least M=M(n,k)>0. A natural question to ask is: What happens for other sets Z? In particular, for finite, but sufficiently dense sets? This question was partially answered in ([16],[20-22]). This study can be naturally related to a certain special settings of the classical Whitney's smooth extension problem. Our goal in the present paper is threefold: first, to provide an overview of the relevant questions and existing results in the general Whitney's problem. Second, we provide an overview of our specific setting and some available results. Third, we provide some new results in our direction. These new results extend the recent result of [21], where an answer to the above question is given via the topological information on Z.