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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]

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 Hölder-continuous functions, but not necessarily smooth. [Copyright &y& Elsevier]

Abstract: The Haar system is an alternative to the classical Fourier bases, being particularly useful for the approximation of discontinuities. The article tackles the construction of a set of fractal functions close to the Haar set. The new system holds the property of constitution of bases of the Lebesgue spaces of -integrable functions on compact intervals. Likewise, the associated fractal series of a continuous function is uniformly convergent. The case owns some peculiarities and is studied separately. [Copyright &y& Elsevier]

Abstract: Fractal Interpolation functions provide natural deterministic approximation of complex phenomena. Cardinal cubic splines are developed through moments (i.e. second derivative of the original function at mesh points). Using tensor product, bicubic spline fractal interpolants are constructed that successfully generalize classical natural bicubic splines. An upper bound of the difference between the natural cubic spline blended fractal interpolant and the original function is deduced. In addition, the convergence of natural bicubic fractal interpolation functions towards the original function providing the data is studied. [Copyright &y& Elsevier]

Abstract: A methodology based on fractal interpolation functions is used in this work to define new real maps on the circle generalizing the classical ones. A partition on the circle and a scale vector enable the modification of the definition and properties of the standard periodic functions. The fractal analogues can be constructed even if the originals are not continuous. The new functions provide Hilbert bases for the square integrable maps on the circle. [Copyright &y& Elsevier]

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]

Abstract: 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. [Copyright &y& Elsevier]