We construct a generalized C r-Fractal Interpolation Function (C r-FIF) f by prescribing any combination of r values of the derivatives f (k) , k = 1, 2,. .. , r, at boundary points of the interval I = [x 0 , x N ]. Our approach to construction settles several questions of Barnsley and Harrington [J. Approx Theory, 57 (1989), pp. 14-34] when construction is not restricted to prescribing the values of f (k) at only the initial endpoint of the interval I. In general, even in the case when r equations involving f (k) (x 0) and f (k) (x N), k = 1, 2,. .. , r, are prescribed, our method of construction of the C r-FIF works equally well. In view of wide ranging applications of the classical cubic splines in several mathematical and engineering problems, the explicit construction of cubic spline FIF f Δ (x) through moments is developed. It is shown that the sequence {f Δ k (x)} converges to the defining data function Φ(x) on two classes of sequences of meshes at least as rapidly as the square of the mesh norm Δ k approaches to zero, provided that Φ (r) (x) is continuous on I for r = 2, 3, or 4.

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Fractal interpolants constructed through iterated function systems prove more general than classical interpolants. In this paper, we assign a family of fractal functions to several classes of real mappings like, for instance, maps defined on sets that are not intervals, maps integrable but not continuous and may be defined on unbounded domains. In particular, based on fractal interpolation functions, we construct fractal Müntz polynomials that successfully generalize classical Müntz polynomials. The parameters of the fractal Müntz system enable the control and modification of the properties of original functions. Furthermore, we deduce fractal versions of classical Müntz theorems. In this way, the fractal methodology generalizes the fundamental sets of the classical approximation theory and we construct complete systems of fractal functions in spaces of continuous and p-integrable mappings on bounded domains.

We construct hidden variable bivariate fractal interpolation surfaces (FIS). The vector valued iterated function system (IFS) is constructed in ℝ4 and its projection in ℝ3 is taken. The extra degree of freedom coming from ℝ4 provides hidden variable, which is an important factor for flexibility and diversity in the interpolated surface. In the present paper, we construct an IFS that generates both self-similar and non-self-similar FIS simultaneously and show that the hidden variable fractal surface may be self-similar under certain conditions.

This paper generalizes the classical spline using a new construction of spline coalescence hidden variable fractal interpolation function (CHFIF). The derivative of a spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of a nondiagonal iterated function system. Our construction generalizes the construction of Barnsley and Harrington (1989), when the construction is not restricted to a particular type of boundary conditions. Spline CHFIFs are likely to be potentially useful in approximation theory due to effects of the hidden variables and these effects are demonstrated through suitable examples in the present work.

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