Binary Operations in Metric Spaces Satisfying Side Inequalities

María A. Navascués, Pasupathi Rajan and Arya Kumar Bedabrata Chand
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María A. Navascués: Departamento de Matemática Aplicada, Escuela de Ingeniería y Arquitectura, Universidad de Zaragoza, 50018 Zaragoza, Spain
Pasupathi Rajan: Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India
Arya Kumar Bedabrata Chand: Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India

Mathematics, 2021, vol. 10, issue 1, 1-17

Abstract: The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was done in the framework of iterated function systems and fractal theory. In this article we extract the main features of this association, and consider binary operations in metric spaces satisfying properties as idempotency and inequalities related to the distance between operated elements with the same right or left factor (side inequalities). Important examples are the logical disjunction and conjunction in the set of integers modulo 2 and the union of compact sets, besides the aforementioned fractal convolution. The operations described are called in the present paper convolutions of two elements of a metric space E . We deduce several properties of these associations, coming from the considered initial conditions. Thereafter, we define self-operators (maps) on E by using the convolution with a fixed component. When E is a Banach or Hilbert space, we add some hypotheses inspired in the fractal convolution of maps, and construct in this way convolved Schauder and Riesz bases, Bessel sequences and frames for the space.

Keywords: metric space; convolution; fractal; Schauder basis; Bessel sequence; Riesz basis; frames (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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