M. A. Navascués

Generalization of Hermite functions by fractal interpolation

Fractal interpolation techniques provide good deterministic representations of complex phenomena. This paper approaches the Hermite interpolation using fractal procedures. This problem prescribes at each support abscissa not only the value of a function but also its first p derivatives. It is shown here that the proposed fractal interpolation function and its first p derivatives are good approximations of the corresponding derivatives of the original function. According to the theorems, the described method allows to interpolate, with arbitrary accuracy, a smooth function with derivatives prescribed on a set of points. The functions solving this problem generalize the Hermite osculatory polynomials.

Smooth fractal interpolation

Fractal methodology provides a general frame for the understanding of real-world phenomena. In particular, the classical methods of real-data interpolation can be generalized by means of fractal techniques. In this paper, we describe a procedure for the construction of smooth fractal functions, with the help of Hermite osculatory polynomials. As a consequence of the process, we generalize any smooth interpolant by means of a family of fractal functions. In particular, the elements of the class can be defined so that the smoothness of the original is preserved. Under some hypotheses, bounds of the interpolation error for function and derivatives are obtained. A set of interpolating mappings associated to a cubic spline is defined and the density of fractal cubic splines in is proven.

FRACTAL BASES OF Lp SPACES

The methodology of fractal sets generates new procedures for the analysis of functions whose graphs have a complex geometric structure. In the present paper, a method for the definition of fractal functions is described. The new mappings are perturbed versions of classical bases as Legendre polynomials, etc. The new elements are non-differentiable and may serve as models for pseudo-random behaviour. The proposed fractal functions have good algebraic properties and good approximation properties as well. In the present paper it is proved that they constitute bases for the most important functional spaces as, for instance, the Lebesgue spaces (1 ≤ p < ∞), where I is a compact interval in the reals.

A relation between fractal dimension and Fourier transform-electroencephalographic study using spectral and fractal parameters

Abstract Fourier parameters display the spectral content of a signal and provide explicit representations in the frequency domain. In the case of electroencephalograms, the FFT algorithm was the only quantifying method used until the 1970s, but this analysis is not conclusive in most pathologies. Fractal techniques provide new tools for the processing of signals whose traces have a sophisticated geometric complexity. The fractal dimension is computed easily in our work by means of explicit formulae involving the time samples. In this paper we obtain, under some hypotheses, an exact relation between the Fourier Transform of a signal and its Fractal Dimension. In the last part of the paper, an electroencephalographic application involving fractal and spectral parameters is described.

FRACTAL TRIGONOMETRIC POLYNOMIALS FOR RESTRICTED RANGE APPROXIMATION

One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions.

FITTING FRACTAL SURFACES ON NON-RECTANGULAR GRIDS

In a complex society, the visualization and interpretation of large amounts of data acquires an increasing importance. This can be done at once by means of two- or three-dimensional maps. To approach this problem, we undertake the construction of several variable fractal functions. In the first place, we introduce real fractal functions defined as perturbations of the classical ones. These basic mappings allow us to compute multidimensional approximations of experimental variables by means of linear combinations of products of fractal functions of Legendre type. The paper proposes a method of non-smooth representation for a large number of three-dimensional data on non-uniform grids. The procedures described are applied in the last part of the paper to the implementation of fitting maps for brain electrical activity.

Numerical integration of affine fractal functions

Abstract This paper studies a method for the numerical integration and representation of functions defined through their samples, when the original “signal” is not explicitly known, but it shows experimentally some kind of self-similarity. In particular, we propose a methodology based on fractal interpolation functions for the computation of the integral that generalize the compound trapezoidal rule. The convergence of the procedure is proved with the only hypothesis of continuity. The rate of convergence is specified in the case of original Holder-continuous functions, but not necessarily smooth.

Time domain indices and discrete power spectrum in electroencephalographic processing

The classical fast Fourier transform (FFT) methods to process experimental signals have the problem of providing multivariant measures with dimension and hence depend on the devices and measure units. These features make the comparisons between different teams and equipments rather difficult. The aim of the present study is to prove the exact mathematical relation between the Hjorth parameters and the spectral powers obtained by means of the quoted algorithm. The proof confirms that these quantifiers can also be used for the exploration of spectral properties of univariant recordings. The formulae proposed provide an easy and low-cost procedure to compute the descriptors if a standard FFT algorithm is performed on the signal. A numerical study of electroencephalographic recordings involving spectral and Hjorth parameters is described in the paper in order to quantify the electroencephalogram of children with problems of attention.

Fractal Polynomials and Maps in Approximation of Continuous Functions

ABSTRACT The primary goal of this article is to establish some approximation properties of fractal functions. More specifically, we establish that a monotone continuous real-valued function can be uniformly approximated with a monotone fractal polynomial, which in addition agrees with the function on an arbitrarily given finite set of points. Furthermore, the simultaneous approximation and \mboxinterpolation which is norm-preserving property of fractal polynomials is established. In the final part of the article, we establish differentiability of a more general class of fractal functions. It is shown that these smooth fractal functions and their derivatives are good approximants for the original function and its \mboxderivatives.

Fractal and Smooth Complexities in Electroencephalographic Processing

The importance of the electroencephalogram (EEG) rests upon the fact that it provides useful information of the normal and pathological brain functions. However, the relations among abnormal EEG, brain functions and disorders are not well known yet. We have proposed numerical quantifiers of the EEG signal, coming from the methodology of fractal mathematics and the theory of approximation. In the first part we describe an alternative to the computation of nonlinear dimensions for this kind of signals. The approach used here is based on a fractal interpolation of the data. In the second part, we describe a method for the computation of smooth complexities based on the interpolation of EEG signals by means of polynomial splines. This kind of functions is used to find quadrature formulas for the spectral moments. Both procedures are applied to treat the electroencephalographic discrimination of a group of children suffering from an Attention Deficit with Hyperactivity Disorder (ADHD).