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Fractal interpolation functions are fixed points of contraction maps on suitable function spaces. In this paper, we introduce the Kantorovich-Bernstein α-fractal operator in the Lebesgue space p(I), 1 ≤ p ≤ ∞. The main aim of this article is to study the convergence of the sequence of Kantorovich-Bernstein fractal functions towards the original functions in p(I) spaces and Lipschitz spaces without affecting the non-linearity of the fractal functions. In the first part of this paper, we introduce a new family of self-referential fractal p(I) functions from a given function in the same space. The existence of a Schauder basis consisting of self-referential functions in p spaces is proven. Further, we derive the fractal analogues of some p(I) approximation results, for example, the fractal version of the classical Müntz-Jackson theorem. The one-sided approximation by the Bernstein α-fractal function is developed. [ABSTRACT FROM AUTHOR]

The objective of the present paper is the study of affine transformations of the plane, which provide self-affine curves as attractors. The properties of these curves depend decisively of the coefficients of the system of affnities involved. The corresponding functions are continuous on a compact interval. If the scale factors are properly chosen one can define Schauder bases ofC[a,b] composed by affine fractal functions close to polygonals. They can be chosen bounded. The basis constants and the biorthogonal sequence of coefficient functionals are studied. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]