An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family. (2nd November 2020)

• Record Type: Journal Article
• Authors: Goldberg, Maxim J.; Kim, Seonja
• Other Names: Kovtunenko Victor Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2020(2020)
• Publication Date: 2020-11-02
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2020/8866826
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : Let X be a topological space equipped with a complete positive σ -finite measure and T a subset of the reals with 0 as an accumulation point. Let a t x, y be a nonnegative measurable function on X × X which integrates to 1 in each variable. For a function f ∈ L 2 X and t ∈ T, define A t f x ≡ ∫ a t x, y f y d y . We assume that A t f converges to f in L 2, as t ⟶ 0 in T . For example, A t is a diffusion semigroup (with T = 0, ∞ ). For W a finite measure space and w ∈ W, select real-valued h w ∈ L 2 X, defined everywhere, with h w L 2 X ≤ 1 . Define the distance D by D x, y ≡ h w x − h w y L 2 W . Our main result is an equivalence between the smoothness of an L 2 X function f (as measured by an L 2 -Lipschitz condition involving a t ·, · and the distance D ) and the rate of convergence of A t f to f .