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Some Properties of the Fractal Convolution of Functions

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Abstract

We consider the fractal convolution of two maps f and g defined on a real interval as a way of generating a new function by means of a suitable iterated function system linked to a partition of the interval. Based on this binary operation, we consider the left and right partial convolutions, and study their properties. Though the operation is not commutative, the one-sided convolutions have similar (but not equal) characteristics. The operators defined by the lateral convolutions are both nonlinear, bi-Lipschitz and homeomorphic. Along with their self-compositions, they are Fréchet differentiable. They are also quasi-isometries under certain conditions of the scale factors of the iterated function system. We also prove some topological properties of the convolution of two sets of functions. In the last part of the paper, we study stability conditions of the dynamical systems associated with the one-sided convolution operators.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada, Escuela de Ingenierí y Arquitectura, Universidad de Zaragoza, Zaragoza, Spain

    María A. Navascués

  2. Department of Mathematics, University of Central Florida, Florida, USA

    Ram N. Mohapatra

  3. Department of Mathematics, Indian Institute of Technology of Madras, Madras, India

    Arya K. B. Chand

Authors
  1. María A. Navascués
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  2. Ram N. Mohapatra
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  3. Arya K. B. Chand
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Correspondence to María A. Navascués.

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Navascués, M.A., Mohapatra, R.N. & Chand, A.K.B. Some Properties of the Fractal Convolution of Functions. Fract Calc Appl Anal 24, 1735–1757 (2021). https://doi.org/10.1515/fca-2021-0075

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  • DOI: https://doi.org/10.1515/fca-2021-0075

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