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A note on the Kolmogorov-type inequalities for more than three norms

Oleg Kovalenko

Abstract

In this note we show that sharp Kolmogorov-type inequalities that estimate the uniform norm $\|f^{(k)}\|$ of the $k$-th derivative of a function $f\colon \mathbb{R}\to\mathbb{R}$ by the values of the uniform norm of $f$ and uniform norms of several its higher derivatives ($\|f^{(r)}\|$ and $\|f^{(r-1)}\|$, or $\|f^{(r)}\|$ and $\|f^{(r-2)}\|$, or $\|f^{(r)}\|$, $\|f^{(r-1)}\|$ and $\|f^{(r-2)}\|$) using standard techniques can be obtained from the known solutions to the Kolmogorov problem about existence of a function with given norms of its derivatives.

A note on the Kolmogorov-type inequalities for more than three norms

Abstract

In this note we show that sharp Kolmogorov-type inequalities that estimate the uniform norm of the -th derivative of a function by the values of the uniform norm of and uniform norms of several its higher derivatives ( and , or and , or , and ) using standard techniques can be obtained from the known solutions to the Kolmogorov problem about existence of a function with given norms of its derivatives.
Paper Structure (5 sections, 4 theorems, 30 equations)

This paper contains 5 sections, 4 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Connection Between the Kolmogorov Problem and the Kolmogorov-Type Inequalities
  3. Extremal Splines
  4. Existence of Extremal Splines
  5. Applications of Theorem \ref{['th::main']}

Key Result

Theorem 1

Let $f\in L^r_{\infty,\infty}$ and $a,\lambda > 0$ be such that $M_0(f) \leq M_0(a\varphi_{\lambda,r})$ and $M_r(f) \leq M_r(a\varphi_{\lambda,r})$. If $\xi,\eta\in\mathbb R$ are such that $f(\xi) = a\varphi_{\lambda,r}(\eta)$, then

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof