Fractal Bases for Banach Spaces of Smooth functions

Navascués, M. A.; Viswanathan, P.; Chand, A. K. B.; Sebastián, María Victoria; Katiyar, S. K.

doi:10.1017/s0004972715000738

Cited by 14 publications

(6 citation statements)

References 16 publications

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“…Substituting x = x 1 + θ (x N − x 1 ), the expression for P * i (θ ) and using the degree elevated form of Q * i (θ ) from Eqn. (18). After some rearrangement, (22) reduces to…”

Section: Rational Cubic Spline Fif Above the Linementioning

confidence: 99%

“…Further, differentiable FIFs [2] initiated a striking relationship between fractal functions and traditional nonrecursive interpolants. Thereafter, many authors have worked in the area of constructing various types of FIFs, including spline FIFs (see, for instance, [5,15,16,17,18,23]) and hidden variable FIFs [3,4,7,13]). In this way, the fractal methodology provides more flexibility and versatility on the choice of interpolant.…”

Section: Introductionmentioning

confidence: 99%

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Parameter Identification of Constrained Data by a New Class of Rational Fractal Function

Katiyar¹,

Chand²,

Jha³

2018

Preprint

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This paper sets a theoretical foundation for the applications of the fractal interpolation functions (FIFs). We construct rational cubic spline FIFs (RCSFIFs) with quadratic denominator involving two shape parameters. The elements of the iterated function system (IFS) in each subinterval are identified befittingly so that the graph of the resulting C 1 -RCSFIF lies within a prescribed rectangle. These parameters include, in particular, conditions on the positivity of the C 1 -RCSFIF. The problem of visualization of constrained data is also addressed when the data is lying above a straight line, the proposed fractal curve is required to lie on the same side of the line. We illustrate our interpolation scheme with some numerical examples.

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“…Substituting x = x 1 + θ (x N − x 1 ), the expression for P * i (θ ) and using the degree elevated form of Q * i (θ ) from Eqn. (18). After some rearrangement, (22) reduces to…”

Section: Rational Cubic Spline Fif Above the Linementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Parameter Identification of Constrained Data by a New Class of Rational Fractal Function

Katiyar¹,

Chand²,

Jha³

2018

Preprint

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“…In [34], Wang and Yu proposed fractal interpolation functions with variable scaling that are suitable to approximate data egenrating function with lesser selfsimilarity. In [22], Navascués et al constructed a k-times continuously differentiable fractal interpolation function with variable scaling and showed the existence of a fractal basis for the space of k-times continuously differentiable functions on the domain of interpolation.…”

Section: Introductionmentioning

confidence: 99%

Zipper Fractal Functions with Variable Scalings

Vijay

Chand

2022

Advances in the Theory of Nonlinear Analysis and Its Application

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Zipper fractal interpolation function (ZFIF) is a generalization of fractal interpolation function through an improved version of iterated function system by using a binary parameter called a signature. The signature allows the horizontal scalings to be negative. ZFIFs have a complex geometric structure, and they can be non-differentiable on a dense subset of an interval I. In this paper, we construct k-times continuously differentiable ZFIFs with variable scaling functions on I. Some properties like the positivity, monotonicity, and convexity of a zipper fractal function and the one-sided approximation for a continuous function by a zipper fractal function are studied. The existence of Schauder basis of zipper fractal functions for the space of k-times continuously differentiable functions and the space of p-integrable functions for p ∈ [1,∞) are studied. We introduce the zipper versions of full Müntz theorem for continuous function and p-integrable functions on I for p ∈ [1,∞).

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“…Fractal functions are not well explored in the field of shape preserving interpolation/approximation. Motivated by theoretical and practical needs, the authors have initiated the study of shape preserving interpolation and approximation using fractal functions [22,23,24,25,26,27,28,29,30,31,32].…”

Section: Introductionmentioning

confidence: 99%

Multivariate Affine Fractal Interpolation

Navascués¹,

Katiyar

Chand

2020

Fractals

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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 [Formula: see text] convergence of this type of interpolants for [Formula: see text] 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 continuous functions defined on a multidimensional compact rectangle is studied.

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Fractal Bases for Banach Spaces of Smooth functions

Cited by 14 publications

References 16 publications

Parameter Identification of Constrained Data by a New Class of Rational Fractal Function

Parameter Identification of Constrained Data by a New Class of Rational Fractal Function

Zipper Fractal Functions with Variable Scalings

Multivariate Affine Fractal Interpolation

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