Fractal perturbation preserving fundamental shapes: Bounds on the scale factors

Full length article: Fractal rational functions and their approximation properties

We prove that the system $\dot{\theta}(t) =\Lambda(t)\theta(t)$
 , $t\in\mathbb{R}_{+}$
 , is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions; we take the exact solutions of the Cauchy problem $\dot{\phi}(t)=\Lambda(t)\phi(t)+e^{i\gamma t}\xi(t)$
 , $t\in\mathbb{R}_{+}$
 , $\phi(0)=\theta_{0}$
 as the approximate solutions of $\dot{\theta}(t)=\Lambda(t)\theta(t)$
 , where γ is any real number, ξ is a 2-periodic, continuous, and bounded vectorial function with $\xi(0)=0$
 , and $\Lambda(t)$
 is a 2-periodic square matrix of order l.

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Connections between Hyers-Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems

A general framework to construct fractal interpolation surfaces (FISs) on rectangular grids was presented and bilinear FIS was deduced by Ruan and Xu (Bull Aust Math Soc 91(3):435–446, 2015). From the view point of operator theory and the stand point of developing some approximation aspects, we revisit the aforementioned construction to obtain a fractal analogue of a prescribed continuous function defined on a rectangular region in $${\mathbb {R}}^2$$. This approach leads to a bounded linear operator analogous to the so-called $$\alpha $$-fractal operator associated with the univariate fractal interpolation function. Several elementary properties of this bivariate fractal operator are reported. We extend the fractal operator to the $${\mathcal {L}}^p$$-spaces for $$1 \le p < \infty $$. Some approximation aspects of the bivariate continuous fractal functions are also discussed.

A Fractal Operator Associated with Bivariate Fractal Interpolation Functions on Rectangular Grids

Fractal interpolation and approximation received a lot of attention in the last thirty years. The main aim of the current article is to study a fractal trigonometric approximants which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. In this paper, we first introduce a new class of fractal approximants, namely, Bernstein $\alpha $
 -fractal functions using the theory of fractal approximation and Bernstein polynomial. Using the proposed class of fractal approximants and imposing no condition on corresponding scaling factors, we establish that the set of Bernstein $\alpha $
 -fractal trigonometric functions is fundamental in the space of continuous periodic functions. Fractal version of Gauss formula of trigonometric interpolation is obtained by means of Bernstein trigonometric fractal polynomials. We study the Bernstein fractal Fourier series of a continuous periodic function $f$
 defined on $[-l,l]$
 . The Bernstein fractal Fourier series converges to $f$
 even if the magnitude of the scaling factors does not approach zero. Existence of the $\mathcal{C}^{r}$
 -Bernstein fractal functions is investigated, and Bernstein cubic spline fractal interpolation functions are proposed based on the theory of $\mathcal{C}^{r}$
 -Bernstein fractal functions.

Bernstein Fractal Trigonometric Approximation

Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of $$\mathcal C ^1$$ - rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form $$\frac{p_i(x)}{q_i(x)},$$ where $$p_i(x)$$ and $$q_i(x)$$ are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in $$\mathcal C ^2$$ is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results.

Shape preservation of scientific data through rational fractal splines

Fractal interpolation provides an efficient way to describe data that have smooth and non-smooth structures. Based on the theory of fractal interpolation functions (FIFs), the Hermite rational cubic spline FIFs (fractal boundary curves) are constructed to approximate an original function along the grid lines of interpolation domain. Then the blending Hermite rational cubic spline fractal interpolation surface (FIS) is generated by using the blending functions with these fractal boundary curves. The convergence of the Hermite rational cubic spline FIS towards an original function is studied. The scaling factors and shape parameters involved in fractal boundary curves are constrained suitably such that these fractal boundary curves are positive whenever the given interpolation data along the grid lines are positive. Our Hermite blending rational cubic spline FIS is positive whenever the corresponding fractal boundary curves are positive. Various collections of fractal boundary curves can be adapted with suitable modifications in the associated scaling parameters or/and shape parameters, and consequently our construction allows interactive alteration in the shape of rational FIS.

Positive blending Hermite rational cubic spline fractal interpolation surfaces

Fractal interpolation is an advanced technique for analysis and synthesis of scientific and engineering data. We introduce the -rational quadratic fractal interpolation functions (FIFs) through a suitable rational quadratic iterated function system (IFS). The novel notion of shape preserving fractal interpolation without any shape parameter is introduced through the rational fractal interpolation model in the literature for the first time. For a prescribed set of monotonic data, we derive the sufficient conditions by restricting the scaling factors for shape preserving -rational quadratic FIFs. A local modification pertaining to any subinterval is possible in this model if the scaling factors are chosen appropriately. We establish the convergence results of a monotonic rational quadratic FIF to the original function in . For given data with derivatives at grids, our approach generates several monotonicity preserving rational quadratic FIFs, whereas this flexibility is not available in the classical approach. Finally, numerical experiments support the importance of the developed rational quadratic IFS scheme through construction of visually pleasing monotonic rational fractal curves including the classical one.

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Monotonicity Preserving Rational Quadratic Fractal Interpolation Functions

In this paper we consider the (inverse) problem of determining the iterated function system (IFS) which produces a shaped fractal interpolant. We develop a new type of rational IFS by using functions of the form , where are cubics and are preassigned quadratics having 3-shape parameters. The fixed point of the developed rational cubic IFS is in , but its derivative varies from a piecewise differentiable function to a continuous nowhere differentiable function. An upper bound of the uniform error between the fixed point of a rational IFS and an original function is deduced for the convergence results. The automatic generations of the scaling factors and shape parameters in the rational IFS are formulated so that its fixed point preserves the positive/monotonic features of prescribed data. The presence of scaling factors provides additional freedom to the shape of the fractal interpolant over its classical counterpart in the modeling of discrete data.

/pdf/rational-iterated-function-system-for-positive-monotonic-48jlch6h9q.pdf

N. Vijender

Papers

Shape preservation of scientific data through rational fractal splines

Bernstein Fractal Trigonometric Approximation

Positive blending Hermite rational cubic spline fractal interpolation surfaces

Monotonicity Preserving Rational Quadratic Fractal Interpolation Functions

Rational iterated function system for positive/monotonic shape preservation