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A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity

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Abstract

We utilize a new definition for the fractional delta operator and prove that it is equivalent by translation to the more commonly used operator. By means of the convolution operation we demonstrate that this new operator is strongly connected to the positivity, monotonicity, and convexity of the functions on which it operates. We also analyze the case of compositions of discrete fractional operators. Finally, since the operator we study here is translationally related to the more commonly used discrete fractional operators, we are able to establish many new results for all types of discrete fractional differences, and we explicitly demonstrate that our results improve all known existing results in the literature.

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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, UNSW Australia, Sydney, NSW, 2052, Australia

    Christopher Goodrich

  2. Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, Santiago, Chile

    Carlos Lizama

Authors
  1. Christopher Goodrich
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  2. Carlos Lizama
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Correspondence to Carlos Lizama.

Additional information

C. Lizama is partially supported by Fondecyt Grant number 1180041.

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Goodrich, C., Lizama, C. A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity. Isr. J. Math. 236, 533–589 (2020). https://doi.org/10.1007/s11856-020-1991-2

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  • DOI: https://doi.org/10.1007/s11856-020-1991-2

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