New equilibria of non-autonomous discrete dynamical systems

In the framework of non-autonomous discrete dynamical systems in metric spaces, we propose new equilibrium points, called quasi-fixed points, and prove that they play a role similar to that of fixed points in autonomous discrete dynamical systems. In this way some sufficient conditions for the convergence of iterative schemes of type [fórmula] in metric spaces are presented, where the maps [fórmula] are contractivities with different fixed points. The results include any reordering of the maps, even with repetitions, and forward and backward directions. We also prove generalizations of the Banach fixed point theorems when the self-map is substituted by a sequence of contractivities with different fixed points. The theory presented links the field of dynamical systems with the theory of iterated function systems. We prove that in some cases the set of quasi-fixed points is an invariant fractal set. The hypotheses relax the usual conditions on the underlying space for the existence of invariant sets in countable iterated function systems

Non-Stationary a-Fractal Surfaces

In this paper, we define non-stationary fractal interpolation surfaces on a rectangular domain and give some upper bounds for their fractal dimension. Next, we define a fractal operator associated with the non-stationary fractal surfaces, and study some properties of it. In particular, we hint at the existence of a Schauder basis consisting of non-stationary fractal functions

Fitting functions of Jackson type for three-dimensional data

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

Concerning the Vector-Valued Fractal Interpolation Functions on the Sierpinski Gasket

The present paper is concerned with the study of vector-valued interpolation functions on the Sierpinski gasket by certain classes of fractal functions. This extends the known results on the real-valued and vector-valued fractal interpolation functions on a compact interval in R and the real-valued fractal interpolation on the Sierpinski gasket. We study the smoothness property of the vector-valued fractal interpolants on the Sierpinski gasket. A few elementary properties of the fractal approximants and the fractal operator that emerge in connection with the vector-valued fractal interpolation on the Sierpinski gasket are indicated. Some constrained approximation aspects of the vector-valued fractal interpolation function on the Sierpinski gasket are pointed out. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG

Cubic spline fractal solutions of two-point boundary value problems with a non-homogeneous nowhere differentiable term

Fractal interpolation functions (FIFs) supplement and subsume all classical interpolants. The major advantage by the use of fractal functions is that they can capture either the irregularity or the smoothness associated with a function. This work proposes the use of cubic spline FIFs through moments for the solutions of a two-point boundary value problem (BVP) involving a complicated non-smooth function in the non-homogeneous second order differential equation. In particular, we have taken a second order linear BVP: y''(x)+Q(x)y'(x)+P(x)y(x)=R(x) with the Dirichlet''s boundary conditions, where P(x) and Q(x) are smooth, but R(x) may be a continuous nowhere differentiable function. Using the discretized version of the differential equation, the moments are computed through a tridiagonal system obtained from the continuity conditions at the internal grids and endpoint conditions by the derivative function. These moments are then used to construct the cubic fractal spline solution of the BVP, where the non-smooth nature of y'' can be captured by fractal methodology. When the scaling factors associated with the fractal spline are taken as zero, the fractal solution reduces to the classical cubic spline solution of the BVP. We prove that the proposed method is convergent based on its truncation error analysis at grid points. Numerical examples are given to support the advantage of the fractal methodology

Multivariate Affine Fractal Interpolation

Fractal interpolation functions capture the irregularity of some data very effectively in comparison with the classical interpolants. They yield a new technique for fitting experimental data sampled from real world signals, which are usually difficult to represent using the classical approaches. The affine fractal interpolants constitute a generalization of the broken line interpolation, which appears as a particular case of the linear self-affine functions for specific values of the scale parameters. We study the p convergence of this type of interpolants for 1 = p < 8 extending in this way the results available in the literature. In the second part, the affine approximants are defined in higher dimensions via product of interpolation spaces, considering rectangular grids in the product intervals. The associate operator of projection is considered. Some properties of the new functions are established and the aforementioned operator on the space of continuousfunctions defined on a multidimensional compact rectangle is studied

Kantorovich-Bernstein a-fractal function in LP spaces

Fractal interpolation functions are fixed points of contraction maps on suitable function spaces. In this paper, we introduce the Kantorovich-Bernstein a-fractal operator in the Lebesgue space Lp(I), 1 = p = 8. The main aim of this article is to study the convergence of the sequence of Kantorovich-Bernstein fractal functions towards the original functions in Lp(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 Lp(I) functions from a given function in the same space. The existence of a Schauder basis consisting of self-referential functions in Lp spaces is proven. Further, we derive the fractal analogues of some Lp(I) approximation results, for example, the fractal version of the classical Müntz-Jackson theorem. The one-sided approximation by the Bernstein a-fractal function is developed

Fractal approximation of Jackson type for periodic phenomena

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 and their classical counterparts are proposed

Fractal Functions of Discontinuous Approximation

A procedure for the definition of discontinuous real functions is developed, based on a fractal methodology. For this purpose, a binary operation in the space of bounded functions on an interval is established. Two functions give rise to a new one, called in the paper fractal convolution of the originals, whose graph is discontinuous and has a fractal structure in general. The new function approximates one of the chosen pair and, under certain conditions, is continuous. The convolution is used for the definition of discontinuous bases of the space of square integrable functions, whose elements are as close to a classical orthonormal system as desired