Now showing 1 - 10 of 35
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    Distribution of Noise in Linear Recurrent Fractal Interpolation Functions for Data Sets with α -Stable Noise
    In this study, we construct a linear recurrent fractal interpolation function (RFIF) with variable scaling parameters for data set with α -stable noise (a generalization of Gaussian noise) on its ordinate, which captures the uncertainty at any missing or unknown intermediate point. The propagation of uncertainty in this linear RFIF is investigated, and a method for estimating parameters of the uncertainty at any interpolated value is provided. Moreover, a simulation study to visualize uncertainty for interpolated values is presented.
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    Binary operations in metric spaces satisfying side inequalities
    (01-01-2022)
    Navascués, María A.
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    Rajan, Pasupathi
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    The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was done in the framework of iterated function systems and fractal theory. In this article we extract the main features of this association, and consider binary operations in metric spaces satisfying properties as idempotency and inequalities related to the distance between operated elements with the same right or left factor (side inequalities). Important examples are the logical disjunction and conjunction in the set of integers modulo 2 and the union of compact sets, besides the aforementioned fractal convolution. The operations described are called in the present paper convolutions of two elements of a metric space E. We deduce several properties of these associations, coming from the considered initial conditions. Thereafter, we define self-operators (maps) on E by using the convolution with a fixed component. When E is a Banach or Hilbert space, we add some hypotheses inspired in the fractal convolution of maps, and construct in this way convolved Schauder and Riesz bases, Bessel sequences and frames for the space.
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    A -Fractal Rational Functions and Their Positivity Aspects
    (01-01-2021)
    Katiyar, S. K.
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    Coalescence hidden variable fractal interpolation function (CHFIF) proves more versatile than classical interpolant and fractal interpolation function (FIF). Using rational functions and CHFIF, a general construction of A-fractal rational functions is introduced for the first time in the literature. This construction of A-fractal rational function also allows us to insert shape parameters for positivity-preserving univariate interpolation. The convergence analysis of the proposed scheme is established. With suitably chosen numerical examples and graphs, the effectiveness of the positivity-preserving interpolation scheme is illustrated.
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    Fractal Convolution Bessel Sequences on Rectangle
    (01-01-2023)
    Pasupathi, R.
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    Navascués, M. A.
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    Fractal functions provide a natural deterministic approximation of complex phenomena and also it has self-similarity. Recently, it has been recognized as an internal binary operation, called fractal convolution. In the present article, we obtain Bessel sequences of L2(I× J) composed of product of fractal convolutions, using the identification of L2(I× J) with the tensor product space L2(I) ⊗ L2(J), where I and J are real compact intervals.
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    Kantorovich-Bernstein α-fractal function in 𠓛P spaces
    (01-02-2020) ;
    Jha, Sangita
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    Navascués, M. A.
    Fractal interpolation functions are fixed points of contraction maps on suitable function spaces. In this paper, we introduce the Kantorovich-Bernstein α-fractal operator in the Lebesgue space 𝓛p(I), 1 ≤ p ≤ ∞. 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 α-fractal function is developed.
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    Shape preserving rational cubic trigonometric fractal interpolation functions
    (01-12-2021)
    Tyada, K. R.
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    Sajid, M.
    This paper is devoted to a hierarchical approach of constructing a class of fractal interpolants with trigonometric basis functions and to preserve the geometric behavior of given univariate data set by these fractal interpolants. In this paper, we propose a new family of C1-rational cubic trigonometric fractal interpolation functions (RCTFIFs) that are the generalized fractal version of the classical rational cubic trigonometric polynomial spline of the form pi(θ)∕qi(θ), where pi(θ) and qi(θ) are cubic trigonometric polynomials with four shape parameters in each sub-interval. The convergence of the RCTFIF towards the original function in C3 is studied. We deduce the simple data dependent sufficient conditions on the scaling factors and shape parameters associated with the C1-RCTFIF so that the proposed RCTFIF lies above a straight line when the interpolation data set is constrained by the same condition. The first derivative of the proposed RCTFIF is irregular in a finite or dense subset of the interpolation interval and matches with the first derivative of the classical rational trigonometric cubic interpolation function whenever all scaling factors are zero. The positive shape preservation is a particular case of the constrained interpolation. We derive sufficient conditions on the trigonometric IFS parameters so that the proposed RCTFIF preserves the monotone or comonotone feature of prescribed data.
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    Positivity Preserving Rational Quartic Spline Zipper Fractal Interpolation Functions
    (01-01-2023)
    Vijay,
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    In this paper, we introduce a class of novel C1 -rational quartic spline zipper fractal interpolation functions (RQS ZFIFs) with variable scalings, where rational spline has a quartic polynomial in the numerator and a cubic polynomial in the denominator with two shape control parameters. We derive an upper bound for the uniform error of the proposed interpolant with a C3 data generating function, and it is shown that our fractal interpolant has O(h2) convergence and can be increased to O(h3) under certain conditions. We restrict the scaling functions and shape control parameters so that the proposed RQS ZFIF is positive, when the given data set is positive. Using this sufficient condition, some numerical examples of positive RQS ZFIFs are presented to support our theory.
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    FAMILY of SHAPE PRESERVING FRACTAL-LIKE BÉZIER CURVES
    (01-09-2020)
    Reddy, K. M.
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    Saravana Kumar, G.
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    Subdivision schemes generate self-similar curves and surfaces for which it has a familiar connection between fractal curves and surfaces generated by iterated function systems (IFS). Overveld [Comput.-Aided Des. 22(9) (1990) 591-597] proved that the subdivision matrices can be perturbated in such a way that it is possible to get fractal-like curves that are perturbated Bézier cubic curves. In this work, we extend the Overveld scheme to nth degree curves, and deduce the condition for curvature continuity and convex hull property. We find the conditions for positive preserving fractal-like Bézier curves in the proposed subdivision matrices. The resulting 2D/3D curves from these binary subdivision matrices resemble with fractal images. Finally, the dependence of the shape of these fractal-like curves on the elements of subdivision matrices is demonstrated with suitably chosen examples.
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    Cyclic iterated function systems
    (01-09-2020)
    Pasupathi, R.
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    Navascués, M. A.
    In this paper, we consider some generalization of the Banach contraction principle, namely cyclic contraction and cyclic φ-contraction. For the application to the fractal, we develop new iterated function systems (IFS) consisting of cyclic contractions and cyclic φ-contractions. Further, we discuss about some special properties of the Hutchinson operator associated with the cyclic (c)-comparison IFS.
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    Fractal Convolution on the Rectangle
    (01-06-2022)
    Pasupathi, R.
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    Navascués, M. A.
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    The primary goal of this article is devoted to the study of fractal bases and fractal frames for L2(I× J) , the collection of all square integrable functions on the rectangle I× J. The fractal function when recognized as an internal binary operation paved way for the construction of right and left partial fractal convolution operators on L2(I) , for any real compact interval I. The aforementioned operators defined on one dimensional space have been applied to obtain operators on the space L2(I× J) by the identification of L2(I× J) with the tensor product space L2(I) ⊗ L2(J). In this paper, we establish properties of this bounded linear operator which eventually helps to prove that L2(I× J) admits Bessel sequences, Riesz bases and frames consisting of products of fractal (self-referential) functions in a nice way.