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An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family

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  • Maxim J. Goldberg
  • Seonja Kim

Abstract

Let be a topological space equipped with a complete positive - finite measure and a subset of the reals with as an accumulation point. Let be a nonnegative measurable function on which integrates to in each variable. For a function and , define . We assume that converges to in , as in . For example, is a diffusion semigroup (with ). For a finite measure space and , select real-valued , defined everywhere, with . Define the distance by . Our main result is an equivalence between the smoothness of an function (as measured by an - Lipschitz condition involving and the distance ) and the rate of convergence of to .

Suggested Citation

  • Maxim J. Goldberg & Seonja Kim, 2020. "An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family," Abstract and Applied Analysis, Hindawi, vol. 2020, pages 1-5, November.
    • Handle: RePEc:hin:jnlaaa:8866826
    DOI: 10.1155/2020/8866826
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