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On discretization of some extremal problems

Oleg Kovalenko

TL;DR

This work advances extremal problems for monotone structures by translating continuous questions into discrete analogues. Through a discretization framework, it derives explicit extremal configurations for both coordinate-wise monotone functions and random processes with monotone trajectories, revealing the underlying combinatorial structure via permutation-based constructions. The main contributions are (i) a discrete minimax characterization for rearrangement-invariant classes, (ii) constructive proofs of extremals in both discrete and continuous settings, and (iii) explicit formulas linking extremals to distribution and rearrangement functions, clarifying the origin of the extreme structures in the continuous problems. The results offer a principled discretization approach that yields sharp bounds and explicit extremals, with implications for approximation theory and related extremal problems on monotone classes.

Abstract

We solve two continuous extremal problems on the classes of monotone functions: in the first problem we find extremal values for a line integral of a coordinate-wise monotone function of two variables from a rearrange\-ment-invariant class of functions; in the second one we find extremal values for the expectation of a random process with monotone trajectories at a random time. In both cases we reduce the continuous problems to their discrete counterparts. The obtained discrete problems are on the one hand interesting on their own, and on the other hand give a natural explanation of the structure of the extremal functions for the continuous problems.

On discretization of some extremal problems

TL;DR

This work advances extremal problems for monotone structures by translating continuous questions into discrete analogues. Through a discretization framework, it derives explicit extremal configurations for both coordinate-wise monotone functions and random processes with monotone trajectories, revealing the underlying combinatorial structure via permutation-based constructions. The main contributions are (i) a discrete minimax characterization for rearrangement-invariant classes, (ii) constructive proofs of extremals in both discrete and continuous settings, and (iii) explicit formulas linking extremals to distribution and rearrangement functions, clarifying the origin of the extreme structures in the continuous problems. The results offer a principled discretization approach that yields sharp bounds and explicit extremals, with implications for approximation theory and related extremal problems on monotone classes.

Abstract

We solve two continuous extremal problems on the classes of monotone functions: in the first problem we find extremal values for a line integral of a coordinate-wise monotone function of two variables from a rearrange\-ment-invariant class of functions; in the second one we find extremal values for the expectation of a random process with monotone trajectories at a random time. In both cases we reduce the continuous problems to their discrete counterparts. The obtained discrete problems are on the one hand interesting on their own, and on the other hand give a natural explanation of the structure of the extremal functions for the continuous problems.
Paper Structure (12 sections, 7 theorems, 87 equations)

This paper contains 12 sections, 7 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Discrete extremal problems
  3. A connection between the maximization and the minimization problems
  4. Auxiliary results
  5. Proof of Theorem \ref{['th::min']}
  6. Proof of Corollary \ref{['th::monotoneNodes']}
  7. Proof of Corollary \ref{['th::emptyIntervalsIntersection']}
  8. Some partial cases
  9. Continuous extremal problems
  10. On discretization of Theorem \ref{['th::coordinatewiseMonotone']}
  11. Proof of Theorem \ref{['th::coordinatewiseMonotone']}
  12. Proof of Theorem \ref{['th::randomProcesses']}

Key Result

Theorem 1

Let $m\colon [0,1]\to[0,1]$ be an increasing bijection. For each continuous non-decreasing function $t\colon [0,1]\to [0,1]$ one has

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Corollary 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof
  • ...and 4 more