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New sharp bounds for the Jacobi heat kernel via an extension of the Dijksma-Koornwinder formula

Adam Nowak, Peter Sjögren, Tomasz Z. Szarek

Abstract

We obtain sharp estimates for the Jacobi heat kernel in a range of parameters where the result has not been established before. This extends and completes an earlier result due to the authors. The proof is based on a generalization of the Dijksma-Koornwinder product formula for Jacobi polynomials.

New sharp bounds for the Jacobi heat kernel via an extension of the Dijksma-Koornwinder formula

Abstract

We obtain sharp estimates for the Jacobi heat kernel in a range of parameters where the result has not been established before. This extends and completes an earlier result due to the authors. The proof is based on a generalization of the Dijksma-Koornwinder product formula for Jacobi polynomials.
Paper Structure (6 sections, 12 theorems, 90 equations)

This paper contains 6 sections, 12 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dijksma-Koornwinder formula
  4. Reduction formula
  5. Preliminaries for the proof of Theorem \ref{['thm:main']}
  6. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem A

Let $\alpha,\beta > -1$ and $T > 0$ be fixed. There exists a constant $C=C(\alpha,\beta,T) > 1$ such that for $\theta,\varphi \in [0,\pi]$ and $0 < t \le T$.

Theorems & Definitions (22)

  • Theorem A
  • Lemma 2.1: NSS0
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Lemma 4.1: NSS2
  • Lemma 4.2
  • ...and 12 more