Papers by Maria Victoria Sebastián
Mathematics
Inspired by the ideas of R. Kannan, we define the new concepts of mutual Kannan contractivity and... more Inspired by the ideas of R. Kannan, we define the new concepts of mutual Kannan contractivity and mutual contractivity between two self-maps on a metric space that generalize the concepts of the Kannan map and contraction. We give some examples and deduce the properties of the operators satisfying this type of condition; in particular, we study the case where the space is normed, and the maps are linear. Then we generalize some theorems proposed by this author on the existence of a fixed point of one operator or a common fixed point for two operators. Our results first prove the existence of a common fixed point of a set of self-maps of any cardinal number (countable or uncountable) satisfying the conditions of Kannan type in metric spaces. The same is proved for a set of maps satisfying the mutual relations of classical contractivity. We prove in both cases the convergence of iterative schemes involving operators with mutual relations of contractivity, proposing sufficient conditio...
Fractal and Fractional
This volume gathers some important advances in the fields of fractional calculus and fractal curv... more This volume gathers some important advances in the fields of fractional calculus and fractal curves and functions [...]
Abstract. The spectral methods (in terms of trigonometric polynomials) are suitable to model peri... more Abstract. The spectral methods (in terms of trigonometric polynomials) are suitable to model periodic or near periodic phenomena. However some experimental variables are far from periodicity. We describe a numerical procedure to obtain trend curves for historic data or short sampled signals, using orthogonal expansions. The approximants chosen are of Legendre type, and perform a low-pass filtering to the data. Additionally, we propose a numerical quantifyer of the variation of the trend curve whose character is non-spectral. In the second part of the text we perform numerical simulations to test the method: the first one is associated with standard models as Gaussian and sinusoidal functions, the second involves a historic record of the Spanish stock reference index IBEX 35.
International Journal of Computer Mathematics, 2018
We study some procedures for the approximation of three-dimensional data on a grid with a hypothe... more We study some procedures for the approximation of three-dimensional data on a grid with a hypothesis of periodicity. The first part proposes a generalization of a discrete periodic approximation defined by Dunham Jackson. The functions used have the advantage of owning an analytical explicit expression in terms of the samples (specific values) of the original function or data. In the second part we describe a continuous approximation function for the same problem, defined through an integral. Some results of the rate of convergence and bounds of the approximation error are presented, with the single hypothesis of Hölder continuity or continuity of the original function.
Fractals, 2018
The reconstruction of an unknown function providing a set of Lagrange data can be approached by m... more The reconstruction of an unknown function providing a set of Lagrange data can be approached by means of fractal interpolation. The power of that methodology allows us to generalize any other interpolant, both smooth and nonsmooth, but the important fact is that this technique provides one of the few methods of nondifferentiable interpolation. In this way, it constitutes a functional model for chaotic processes. This paper studies a generalization of an approximation formula proposed by Dunham Jackson, where a wider range of values of an exponent of the basic trigonometric functions is considered. The trigonometric polynomials are then transformed in close fractal functions that, in general, are not smooth. For suitable election of this parameter, one obtains better conditions of convergence than in the classical case: the hypothesis of continuity alone is enough to ensure the convergence when the sampling frequency is increased. Finally, bounds of discrete fractal Jackson operators...
Computational and Mathematical Methods, 2019
In this paper, we establish a new formula which generalizes the Jackson trigonometric interpolati... more In this paper, we establish a new formula which generalizes the Jackson trigonometric interpolation for a 2 -periodic function. This generalization is done by using various positive exponents in the basic nodal functions that gives a wide variety of bases during approximation. For a Hölder continuous periodic function, we compute the uniform interpolation error bound of the corresponding generalized Jackson interpolant, and prove the convergence of the proposed interpolant. We also show that the mentioned approximation procedure is stable. In the last part, we consider a family of fractal interpolants associated with the generalized Jackson approximation functions under discussion.
Journal of Computational and Applied Mathematics, 2019
This article proposes a generalization of the Fourier interpolation formula, where a wider range ... more This article proposes a generalization of the Fourier interpolation formula, where a wider range of the basic trigonometric functions is considered. The extension of the procedure is done in two ways: adding an exponent to the maps involved, and considering a family of fractal functions that contains the standard case. The studied interpolation converges for every continuous function, for a large range of the nodal mappings chosen. The error of interpolation is bounded in two ways: one theorem studies the convergence for Hölder continuous functions and other develops the case of merely continuous maps. The stability of the approximation procedure is proved as well.
Mathematical Methods in the Applied Sciences, 2014
The record of the oscillations of the electric potential of the human brain provides useful infor... more The record of the oscillations of the electric potential of the human brain provides useful information about the mind activity at rest and during the achievement of sensory and cognitive processing tasks. The use of appropriate quantitative tools assigning numerical values to the observed variable is necessary to define good descriptors of the electroencephalogram, allowing comparisons between different recordings. In this line, we propose a numerical method for the spectral and temporal reconstruction of a brain signal. The convergence of the procedure is analyzed, providing results of the concerned approximation error. In a second part of the text, we use the methodology described to the quantification of the bioelectric variations produced in the brain waves for the execution of a test of attention related to military simulation. Copyright © 2014 JohnWiley & Sons, Ltd.
Applied Mathematics, 2014
The object of this short survey is to revive interest in the technique of fractal interpolation. ... more The object of this short survey is to revive interest in the technique of fractal interpolation. In order to attract the attention of numerical analysts, or rather scientific community of researchers applying various approximation techniques, the article is interspersed with comparison of fractal interpolation functions and diverse conventional interpolation schemes. There are multitudes of interpolation methods using several families of functions: polynomial, exponential, rational, trigonometric and splines to name a few. But it should be noted that all these conventional nonrecursive methods produce interpolants that are differentiable a number of times except possibly at a finite set of points. One of the goals of the paper is the definition of interpolants which are not smooth, and likely they are nowhere differentiable. They are defined by means of an appropriate iterated function system. Their appearance fills the gap of non-smooth methods in the field of approximation. Another interesting topic is that, if one chooses the elements of the iterated function system suitably, the resulting fractal curve may be close to classical mathematical functions like polynomials, exponentials, etc. The authors review many results obtained in this field so far, although the article does not claim any completeness. Theory as well as applications concerning this new topic published in the last decade are discussed. The one dimensional case is only considered.
Mathematical Inequalities & Applications, 2006
Journal of Approximation Theory, 2004
emis.ams.org
Fractal interpolation functions provide new methods for approximating experimental data. The main... more Fractal interpolation functions provide new methods for approximating experimental data. The main difference with the classical procedures is the fractal character of the curves used to fit the points. In the present paper, an affine fractal interpolation technique is applied to find an explicit formula to obtain the energy of a real periodic evenly sampled signal. Under some hypothesis about the original function, an error bound is given.
Contributions to Statistics, 2016
In Chapter 9 titled "Stock Indices in Emerging and Consolidated Economies from a Fractal Perspect... more In Chapter 9 titled "Stock Indices in Emerging and Consolidated Economies from a Fractal Perspective", the name of the author was incorrect. The name should read Mario Latorre instead of Miguel Latorre.
Stock Indices in Emerging and Consolidated Economies from a Fractal Perspective
Contributions to Statistics, 2016
In this chapter we analyze five international stock indices (Eurostoxx 50, Ibovespa, Nikkei 225, ... more In this chapter we analyze five international stock indices (Eurostoxx 50, Ibovespa, Nikkei 225, Sensex, and Standard & Poor’s 500, of Europe, Brazil, Japan, India, and USA, respectively) in order to check and measure their geometric complexity. From the financial point of view, we look for numerical differences in the wave patterns between emerging and consolidated economies. We are concerned with the discrimination of new and old markets, in a self-similar perspective. From the theoretical side, we wish to seek evidences pointing to a fractal structure of the daily closing prices. We wish to inquire about the type of randomness in the movement and evolution of the indices through different tests. Specifically, we use several procedures to find the suitability of an exponential law for the spectral power, in order to determine if the indices admit a model of colored noise and, in particular, a Brownian random pattern. Further, we check a possible structure of fractional Brownian motion as defined by Mandelbrot. For it, we determine several parameters from the spectral field and the fractal theory that quantify the values of the stock records and its trends.
Mathematical Inequalities & Applications, 2005
The approximation of experimental data can be envisaged in the light of fractal interpolation fun... more The approximation of experimental data can be envisaged in the light of fractal interpolation functions defined by iterated function systems. In the particular case of polynomial fractal interpolation functions, the method can be considered as a generalization of the splines of the same kind. That extension is verified under preservation of the smoothness of the function. A bound of the interpolation error by odd degree polynomial fractal interpolation functions is obtained here. The upper limit is also given for high-order derivatives, up to the (2m − 2)th derivative if the polynomials are of degree 2m − 1. The result can be considered a fractal version of Schultz's theorem for odd degree polynomial splines.
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Papers by Maria Victoria Sebastián