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Extending Theorems of Boros and Menzer

Marek Balcerzak, Michał Popławski

Abstract

We extend results of Boros and Menzer on the alternative equation $f(x)f(y)=0$ for generalized polynomials $f$, and their theorems on the conditional inequality $f(x)f(y)\ge 0$ for generalized monomials $f$ of even degree. We use similar methods and ideas. We replace the largeness, of the respective Borel plane set $D$, in the measure or in the Baire category sense, by its largeness in the mixed measure-category sense.

Extending Theorems of Boros and Menzer

Abstract

We extend results of Boros and Menzer on the alternative equation for generalized polynomials , and their theorems on the conditional inequality for generalized monomials of even degree. We use similar methods and ideas. We replace the largeness, of the respective Borel plane set , in the measure or in the Baire category sense, by its largeness in the mixed measure-category sense.

Paper Structure

This paper contains 5 sections, 10 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. The Fubini products of $\sigma$-ideals
  3. A theorem on the alernative equation
  4. A theorem on the conditional inequality
  5. Concluding remarks

Key Result

Theorem 1.1

(See BM) Let $f\colon\mathbb R\to\mathbb C$ be a generalized polynomial such that $f(x)f(y)=0$ for all $(x,y)\in D$ where $D\subseteq\mathbb R^2$. Assume that one of the conditions is true: Then $f(x)=0$ for each $x\in\mathbb R$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1
  • proof
  • Corollary 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • Lemma 4.1
  • ...and 6 more