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Lower bounds for high derivatives of smooth functions with given zeros

Gil Goldman, Yosef Yomdin

Abstract

Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} |f(z)|=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty interior, then for each $k=1,\ldots,d+1$ 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 $Z$.

Lower bounds for high derivatives of smooth functions with given zeros

Abstract

Let be a times continuously differentiable function on the unit ball , with . A well-known fact is that if vanishes on a set with a non-empty interior, then for each the norm of the -th derivative is at least . A natural question to ask is: What happens for other sets ? 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 .
Paper Structure (9 sections, 10 theorems, 15 equations)

This paper contains 9 sections, 10 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Whitney's smooth extension problem
  3. Smooth rigidity
  4. Rigidity and Remez constant of $Z$
  5. Rigidity and test curves
  6. Polynomial test curves of higher degree
  7. Rigidity via the topology of $Z$
  8. Rigidity via singular points and values
  9. New results

Key Result

proposition thmcounterproposition

For any $Z\subset B^1$ we have ${\cal RG}_d(Z)\ge \frac{(d+1)!}{2^{d+1}},$ if $Z$ consists of at least $d+1$ different points, and ${\cal RG}_d(Z)=0$ if $Z$ consists of at most $d$ different points.

Theorems & Definitions (11)

  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • definition thmcounterdefinition
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • corollary thmcountercorollary
  • corollary thmcountercorollary
  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • ...and 1 more