Fundamental Sets of Fractal Functions

Jha

Chand

et al. 2019

Comp and Math Methods

In this paper, we establish a new formula that generalizes the Jackson trigonometric interpolation for a 2π‐periodic function. This generalization is done by using various positive exponents in the basic nodal functions that gives a wide variety of bases during approximation. For a Hölder continuous periodic function, we compute the uniform interpolation error bound of the corresponding generalized Jackson interpolant and prove the convergence of the proposed interpolant. We also show that the mentioned approximation procedure is stable. In the last part, we consider a family of fractal interpolants associated with the generalized Jackson approximation functions under discussion.

Section: Ifs Theory and -Fractal Functionsmentioning

confidence: 99%

A fractal class of generalized Jackson interpolants

Jha

Chand

et al. 2019

Comp and Math Methods

“…Consider the vector space of continuous functions C(I ) endowed with the uniform norm f ∞ = sup{|f (t)| : t ∈ I }. The following inequalities are proved in [7]:…”

Section: Propositionmentioning

confidence: 99%

Fractal Interpolation on a Torus

2008

Acta Appl Math

Self Cite

A very general method of fractal interpolation on T 1 is proposed in the first place. The approach includes the classical cases using trigonometric functions, periodic splines, etc. but, at the same time, adds a diversity of fractal elements which may be more appropriate to model the complexity of some variables. Upper bounds of the committed error are provided. The arguments avoid the use of derivatives in order to handle a wider framework. The Lebesgue constant of the associated partition plays a key role. The procedure is proved convergent for the interpolation of specific functions with respect to some nodal bases. In a second part, the approximation is then extended to bidimensional tori via tensor product of interpolation spaces. Some sufficient conditions for the convergence of the process in the Fourier case are deduced.

“…Navascués and Chand [14] extended the α-fractal function to the setting of L p -spaces using the standard density argument. In this construction of fractal functions in L p -spaces, the following assumptions are inevitable: (i) base function b = Lf , where L is a bounded linear map on L p (I) (ii) scaling vector satisfies |α| ∞ < 1 (iii) the exponent p is such that 1 ≤ p < ∞.…”

Section: Introductionmentioning

confidence: 99%

Associate fractal functions inLp-spaces and in one-sided uniform approximation

Viswanathan

Journal of Mathematical Analysis and Applications

Chand

2016

Fractal interpolation function defined with the aid of iterated function system can be employed to show that any continuous real-valued function defined on a compact interval is a special case of a class of fractal functions (self-referential functions). Elements of the iterated function system can be selected appropriately so that the corresponding fractal function enjoys certain properties. In the first part of the paper, we associate a class of self-referential L p -functions with a prescribed L p -function. Further, we apply our construction of fractal functions in L p -spaces in some approximation problems, for instance, to derive fractal versions of the full Müntz theorems in L p -spaces. The second part of the paper is devoted to identify parameters so that the fractal functions affiliated to a given continuous function satisfy certain conditions, which in turn facilitate them to find applications in some one-sided uniform approximation problems.