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On the $L^1$ and pointwise divergence of continuous functions

Karol Gryszka, Paweł Pasteczka

Abstract

For a family of continuous functions $f_1,f_2,\dots \colon I \to \mathbb{R}$ ($I$ is a fixed interval) with $f_1\le f_2\le \dots$ define a set $$ I_f:=\big\{x \in I \colon \lim_{n \to \infty} f_n(x)=+\infty\big\}.$$ We study the properties of the family of all admissible $I_f$-s and the family of all admissible $I_f$-s under the additional assumption $$ \lim_{n \to \infty} \int_x^y f_n(t)\:dt=+\infty \quad \text{ for all }x,y \in I\text{ with }x<y.$$ The origin of this problem is the limit behaviour of quasiarithmetic means.

On the $L^1$ and pointwise divergence of continuous functions

Abstract

For a family of continuous functions ( is a fixed interval) with define a set We study the properties of the family of all admissible -s and the family of all admissible -s under the additional assumption The origin of this problem is the limit behaviour of quasiarithmetic means.

Paper Structure

This paper contains 6 sections, 13 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Rephrasing of the problem
  3. General properties of $\Omega(I)$ and $\Omega_0(I)$
  4. Max-families with noninteger Hausdorff dimension
  5. Cantor-type construction
  6. Jarnık-type construction

Key Result

Proposition 1

Let $X\subset I$ be an arbitrary set. If $\lambda(X \cap J)=0$ for some open interval $J$, then there exists a sequence $(f_n \colon I \to \mathbb{R})_{n=1}^\infty$ of continuous functions with $f_1 \le f_2\le\dots$ which is not a max-family, although $I_f \supset X$.

Figures (1)

  • Figure 1: Construction of $D_n$'s.

Theorems & Definitions (24)

  • Proposition 1: Pas16a, Proposition 4.1
  • Proposition 2: Pas16a, Proposition 4.2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • Remark 1
  • proof
  • Lemma 3
  • ...and 14 more