[Paper Review] The best simultaneous approximation in linear 2-normed spaces

Mehmet Açıkgöz|arXiv (Cornell University)|May 15, 2012

Fixed Point Theorems Analysis17 references3 citations

TL;DR

This paper introduces a novel framework for best simultaneous approximation in linear 2-normed spaces by extending classical approximation concepts to 2-normed settings. It establishes the existence and uniqueness of best simultaneous approximations under strict convexity and compactness conditions, proving that a unique minimizer exists for the supremum of 2-norms to a fixed vector, generalizing classical results to 2-normed structures.

ABSTRACT

In this paper, we shall investigate and analyse a new study on the best simultaneous approximation in the context of linear 2-normed spaces inspired by Elumalai and his coworkers in Elumalai. The basis of this investigation is to extend and refinement the definition of the classical aproximation, best approximation and some related concepts to linear 2-normed spaces.

Motivation & Objective

• To extend classical best approximation theory to the context of linear 2-normed spaces.
• To define and analyze simultaneous best approximation in 2-normed spaces using a new formulation based on the supremum of 2-norms.
• To establish conditions under which a unique best simultaneous approximation exists.
• To generalize results from Banach space approximation theory to 2-Banach and 2-normed spaces.
• To provide foundational definitions and properties for 2-normed spaces and their approximation theory.

Proposed method

• Proposes a new definition of best simultaneous approximation in linear 2-normed spaces using the supremum of 2-norms to a fixed vector.
• Applies the concept of distance in 2-normed spaces via the infimum of ||x - g, z|| over g in a subspace G.
• Uses properties of strict convexity and compactness in the subspace Y to ensure uniqueness of the minimizer.
• Employs the parallelogram-type inequality in 2-norms to prove uniqueness, showing ||y₀ - y₀′, z||² ≤ 0 implies y₀ = y₀′.
• Derives lemmas and propositions to support the existence and uniqueness of the best simultaneous approximation.
• Applies the infimum and limit processes on Cauchy sequences to show convergence to the optimal approximation.

Experimental results

Research questions

• RQ1 Can the classical notion of best approximation be meaningfully extended to linear 2-normed spaces?
• RQ2 Under what conditions does a unique best simultaneous approximation exist in a 2-normed space?
• RQ3 How does the supremum of 2-norms to a fixed vector influence the existence and uniqueness of simultaneous approximations?
• RQ4 What role does strict convexity play in ensuring uniqueness of the best simultaneous approximation in 2-normed spaces?
• RQ5 How can 2-functional and subspace structures be used to define and solve simultaneous approximation problems in 2-normed spaces?

Key findings

• A unique best simultaneous approximation exists in a linear 2-normed space when the approximating set Y is compact and the space is strictly convex.
• The infimum of ||x₀ - y, z|| over y ∈ Y is achieved at a unique point y₀ ∈ Y for x₀ ∉ Y and z linearly independent from elements of G.
• The proof of uniqueness relies on a 2-norm parallelogram-type inequality, showing that ||y₀ - y₀′, z||² ≤ 0, which implies y₀ = y₀′.
• The existence of a best simultaneous approximation is guaranteed under the given conditions, extending classical results to 2-normed settings.
• The supremum formulation of the approximation error ensures robustness in the presence of multiple functions to approximate.
• The result holds for linearly independent f₁, f₂, and b in X, ensuring non-degenerate 2-norm behavior.