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Bernstein inequality on parabolic domains

Yuan Xu

Abstract

Several families of sharp Bernstein inequalities are established on the weighted $L^2$ space over parabolic domains, which include bounded or unbounded rotational paraboloids and parabolic surfaces. The main tool is a second-order differential operator satisfied by a specific basis of orthogonal polynomials in weighted $L^2$ space.

Bernstein inequality on parabolic domains

Abstract

Several families of sharp Bernstein inequalities are established on the weighted space over parabolic domains, which include bounded or unbounded rotational paraboloids and parabolic surfaces. The main tool is a second-order differential operator satisfied by a specific basis of orthogonal polynomials in weighted space.

Paper Structure

This paper contains 8 sections, 22 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminary: sharp $L^2$ Bernstein inequalities
  3. Bernstein inequalities on paraboloids
  4. Jacobi polynomials on bounded paraboloid
  5. Laguree polynomials on unbounded paraboloid
  6. Bernstein inequalities on parabolic surfaces
  7. Jacobi polynomials on bounded parabolic surface
  8. Laguerre polynomials on unbounded parabolic surface

Key Result

Lemma 2.1

Assume the spectral operator ${\mathfrak D}$ exists. For $f\in L^2(\Omega, W)$, In particular, if $f \in \Pi_n(\Omega)$, then $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (30)

  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 20 more