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.
Highlights
Let ( Z2, d) be a metric space defined as in the above example and ∧ be the logical conjunction on E defined by 0 ∧ 0 = 0, 0 ∧ 1 = 0, 1 ∧ 0 = 0, 1 ∧ 1 = 1, ∧ is a commutative convolution operator on E; Fractal convolution of functions: In many real-world phenomena, such as signal and image processing, economic and climatic series, bioelectric recordings, cartography, etc., the scientists need to manage irregular and sharp forms, and the standard functions often fall short to approach them
We consider binary operations in metric spaces satisfying two inequalities related to the metric
Remarkable particular cases are the logical conjunction and disjunction, the union of compact sets and the fractal convolution of functions that we proposed in previous papers
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. We 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). It owns other properties that will be described in terms of the distance between operated and original elements These properties are inherited by the convolution subsets. Of the model as zero, one obtains the component function f In this sense, the fractal convolution provides a family of mappings that contains the original. Some authors generalize the concept of distance to maps of type d : X × X → G, where G owns an algebraic structure composed of binary operations and/or relations ([8]).
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