Local Fractal Functions and Function Spaces

In this paper we define an internal binary operation between functions called in the text fractal convolution, that applies a pair of mappings into a fractal function. This is done by means of a suitable Iterated Function System. We study in detail the operation in L p spaces and in sets of continuous functions, in a different way to previous works of the authors. We develop some properties of the operation and its associated sets. The lateral convolutions with the null function provide linear operators whose characteristics are explored. The last part of the article deals with the construction of convolved fractals bases and frames in Banach and Hilbert spaces of functions.

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“…The next result is a special case of Theorem 1 in [16] and its validity follows directly from the above considerations.…”

Section: A Fractal-type Operation Of Fractal Functionsmentioning

confidence: 73%

Fractal Convolution: A New Operation Between Functions

Navascués

Massopust

2019

FCAA

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“…The research reported here is admittedly influenced to an extent by works on fractal functions and local fractal functions by Massopust; see [6,7]. However, it should be noted that our exposition aims at a different goal.…”

Section: Introductionmentioning

confidence: 93%

A Fractal Operator on Some Standard Spaces of Functions

Viswanathan

Navascués

2017

Proceedings of the Edinburgh Mathematical Society

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By appropriate choices of elements in the underlying iterated function system, methodology of fractal interpolation entitles one to associate a family of continuous selfreferential functions with a prescribed real-valued continuous function on a real compact interval. This procedure elicits what is referred to as α-fractal operator on C(I), the space of all real-valued continuous functions defined on a compact interval I. With an eye towards connecting fractal functions with other branches of mathematics, in this article, we continue to investigate fractal operator in more general spaces such as the space B(I) of all bounded functions and Lebesgue space L p (I), and some standard spaces of smooth functions such as the space C k (I) of k-times continuously differentiable functions, Hölder spaces C k,σ (I), and Sobolev spaces W k,p (I). Using properties of the α-fractal operator, the existence of Schauder basis consisting of self-referential functions is established.

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“…For 0 < p < 1, f p defines only a quasi-norm on L p (I) and L p (I) is complete with respect to the metric d p (f, g) := f − g p p . References [10,11] provided us with basic tools which we have modified and adapted to prove:…”

Section: Fractal Functions In L P -Spaces Revisitedmentioning

confidence: 99%

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

Viswanathan

Navascués

Chand

2016

Journal of Mathematical Analysis and Applications

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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.

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Local Fractal Functions and Function Spaces

Cited by 14 publications

References 9 publications

Fractal Convolution: A New Operation Between Functions

Fractal Convolution: A New Operation Between Functions

A Fractal Operator on Some Standard Spaces of Functions

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

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