An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family
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 atx,y be a nonnegative measurable function on X×X which integrates to 1 in each variable. For a function f∈L2X and t∈T, define Atfx≡∫ atx,yfy dy. We assume...
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Wiley
2020-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2020/8866826 |
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| author | Maxim J. Goldberg Seonja Kim |
| author_facet | Maxim J. Goldberg Seonja Kim |
| author_sort | Maxim J. Goldberg |
| collection | DOAJ |
| description | 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 atx,y be a nonnegative measurable function on X×X which integrates to 1 in each variable. For a function f∈L2X and t∈T, define Atfx≡∫ atx,yfy dy. We assume that Atf converges to f in L2, as t⟶0 in T. For example, At is a diffusion semigroup (with T=0,∞). For W a finite measure space and w∈W, select real-valued hw∈L2X, defined everywhere, with hwL2X≤1. Define the distance D by Dx,y≡hwx−hwyL2W. Our main result is an equivalence between the smoothness of an L2X function f (as measured by an L2-Lipschitz condition involving at·,· and the distance D) and the rate of convergence of Atf to f. |
| format | Article |
| id | doaj-art-3125aa4677b34cabb535a5f8fb9d4a2a |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-3125aa4677b34cabb535a5f8fb9d4a2a2025-02-03T06:46:59ZengWileyAbstract and Applied Analysis1085-33751687-04092020-01-01202010.1155/2020/88668268866826An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator FamilyMaxim J. Goldberg0Seonja Kim1Theoretical and Applied Science, Ramapo College of New Jersey, Mahwah, NJ 07430, USAMathematics Department, Middlesex College, Edison, NJ 08818, USALet 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 atx,y be a nonnegative measurable function on X×X which integrates to 1 in each variable. For a function f∈L2X and t∈T, define Atfx≡∫ atx,yfy dy. We assume that Atf converges to f in L2, as t⟶0 in T. For example, At is a diffusion semigroup (with T=0,∞). For W a finite measure space and w∈W, select real-valued hw∈L2X, defined everywhere, with hwL2X≤1. Define the distance D by Dx,y≡hwx−hwyL2W. Our main result is an equivalence between the smoothness of an L2X function f (as measured by an L2-Lipschitz condition involving at·,· and the distance D) and the rate of convergence of Atf to f.http://dx.doi.org/10.1155/2020/8866826 |
| spellingShingle | Maxim J. Goldberg Seonja Kim An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family Abstract and Applied Analysis |
| title | An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family |
| title_full | An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family |
| title_fullStr | An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family |
| title_full_unstemmed | An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family |
| title_short | An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family |
| title_sort | equivalence between the limit smoothness and the rate of convergence for a general contraction operator family |
| url | http://dx.doi.org/10.1155/2020/8866826 |
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