[Paper Review] Consequences of the product rule in Stieltjes differentiability

Francisco J. Fernández, Ignacio Márquez Albés+1 more|arXiv (Cornell University)|May 20, 2022

Functional Equations Stability Results4 citations

TL;DR

This paper investigates the higher-order differentiability of products of functions under Stieltjes differentiation, deriving a modified product rule that includes additional terms involving the jump function Δg and the point shift t*. The key contribution is a necessary and sufficient condition for the g-continuity of the first-order Stieltjes derivative of the product, which depends on the behavior of the jump function Δg and the functions' derivatives at t*.

ABSTRACT

This work revolves around the study of differentiability in the Stieltjes sense of a product of functions. A formula for the first order derivative has been obtained in the past, which is similar to the usual one with some extra terms in its expression. The aim of this paper is to take this behavior into account to study under which conditions we can guarantee the existence of higher order derivatives, while obtaining some other interesting results for the Stieltjes derivative along the way. We also investigate the regularity of the product of two functions.

Motivation & Objective

• To investigate the conditions under which the product of two g-differentiable functions admits higher-order Stieltjes derivatives.
• To analyze the role of the point shift t* and the jump function Δg in the product rule for Stieltjes derivatives.
• To determine when the Stieltjes derivative of the product remains g-continuous, particularly focusing on the behavior of Δg* and the functions evaluated at t*.
• To establish necessary and sufficient conditions for the product of two functions in 𝒞g¹ to remain in the same class, addressing the failure of the standard algebraic structure under product.
• To clarify the differentiability properties of Δg* and f* (f evaluated at t*) in the context of Stieltjes calculus, which are critical for higher-order analysis.

Proposed method

• Derives a generalized product rule for Stieltjes derivatives: (f₁f₂)′_g(t) = f′₁(t)f₂(t*) + f′₂(t)f₁(t*) + f′₁(t)f′₂(t)Δg(t*), where t* depends on t.
• Introduces the concept of f* (f evaluated at t*) and analyzes its g-differentiability, showing that f* is g-differentiable only under specific regularity conditions.
• Studies the g-differentiability of Δg* (the jump function evaluated at t*) and provides counterexamples to show that Δg* is not always g-differentiable.
• Uses measure-theoretic tools, including μg-integrability and g-a.e. convergence, to analyze the continuity and integrability of derivative terms.
• Applies Lemma 2.14 on g-continuity and the finiteness of ∑Δg(s) over compact sets to prove the main continuity result.
• Employs contradiction arguments involving sequences {tₙ} with g(tₙ)→g(t) and |h(tₙ)|≥ε₀ to prove the necessity of the condition ∑Δg(s) < ∞ for g-continuity of the derivative term.

Experimental results

Research questions

• RQ1 Under what conditions is the product of two g-differentiable functions twice g-differentiable?
• RQ2 When is the function f* (f evaluated at t*) g-differentiable at a point t?
• RQ3 When is the jump function Δg* g-differentiable, and what role does it play in the product rule?
• RQ4 What is the necessary and sufficient condition for the g-continuity of the first-order Stieltjes derivative of the product?
• RQ5 How does the behavior of Δg on Dg affect the regularity of the product’s derivative?

Key findings

• The first-order Stieltjes derivative of the product f₁f₂ is given by a modified product rule involving f₂(t*), f₁(t*), and Δg(t*), which introduces new terms absent in classical calculus.
• The function f* (f evaluated at t*) is g-differentiable at t only if f is continuous at t* and f′_g(t) = 0, which is a restrictive condition.
• The jump function Δg* is not necessarily g-differentiable; counterexamples show that even if g is nondecreasing and left-continuous, Δg* may fail to be g-differentiable.
• The g-continuity of (f₁f₂)′_g at t ∈ ((a*,b) ∩ A_g) \ H_g holds if and only if lim_{s→t} (f′₁(s)f′₂(s)Δg(s)) = 0, which is equivalent to the condition that ∑_{s∈[a,b]∩D_g} Δg(s) < ∞.
• The condition ∑_{s∈[a,b]∩D_g} Δg(s) < ∞ is both necessary and sufficient for the g-continuity of the derivative term h = f′₁f′₂Δg*.
• The result generalizes [4, Proposition 3.17], providing a necessary and sufficient condition without requiring one function to be classically continuous.