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Journal articles on the topic 'Fractal interpolation functions'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Al-Jawfi, Rashad A. "3D Fractal Interpolation Functions." Nanoscience and Nanotechnology Letters 12, no. 1 (January 1, 2020): 120–23. http://dx.doi.org/10.1166/nnl.2020.3081.

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For visualizing one, two- and/or three-dimensional data, reconstruction mechanism is generally considered. In the present study, I have examined the application of iterated function systems in interpolation. I have also presented new fractal interpolation derivations for three-dimensional scalar data. The interpolations can indicate uncertainty of the data, allow tenability from the data, represent the data statistically at different scales, and may also more accurately facilitate data analysis.

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2

Bouboulis, P., and L. Dalla. "Fractal Interpolation Surfaces derived from Fractal Interpolation Functions." Journal of Mathematical Analysis and Applications 336, no. 2 (December 2007): 919–36. http://dx.doi.org/10.1016/j.jmaa.2007.01.112.

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3

Barnsley, Michael F. "Fractal functions and interpolation." Constructive Approximation 2, no. 1 (December 1986): 303–29. http://dx.doi.org/10.1007/bf01893434.

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4

Barnsley, M. F., J. Elton, D. Hardin, and P. Massopust. "Hidden Variable Fractal Interpolation Functions." SIAM Journal on Mathematical Analysis 20, no. 5 (September 1989): 1218–42. http://dx.doi.org/10.1137/0520080.

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5

DENİZ, Ali, and Yunus ÖZDEMİR. "Graph-directed fractal interpolation functions." TURKISH JOURNAL OF MATHEMATICS 41 (2017): 829–40. http://dx.doi.org/10.3906/mat-1604-39.

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6

Chand, A. K. B., N. Vijender, P. Viswanathan, and A. V. Tetenov. "Affine zipper fractal interpolation functions." BIT Numerical Mathematics 60, no. 2 (September 6, 2019): 319–44. http://dx.doi.org/10.1007/s10543-019-00774-3.

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7

Drakopoulos, V., and N. Vijender. "Univariable affine fractal interpolation functions." Theoretical and Mathematical Physics 207, no. 3 (June 2021): 689–700. http://dx.doi.org/10.1134/s0040577921060015.

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8

KAPOOR, G. P., and SRIJANANI ANURAG PRASAD. "CUBIC SPLINE SUPER FRACTAL INTERPOLATION FUNCTIONS." Fractals 22, no. 01n02 (March 2014): 1450005. http://dx.doi.org/10.1142/s0218348x14500054.

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In the present work, the notion of Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated. It is shown that, for an equidistant partition points of [x0, xN], the interpolating Cubic Spline SFIF[Formula: see text] and their derivatives [Formula: see text] converge respectively to the data generating function y(x) ≡ y(0)(x) and its derivatives y(j)(x) at the rate of h2-j+ϵ(0 < ϵ < 1), j = 0, 1, 2, as the norm h of the partition of [x0, xN] approaches zero. The convergence results for Cubic Spline SFIF found here show that any desired accuracy can be achieved in the approximation of a regular data generating function and its derivatives by a Cubic Spline SFIF and its corresponding derivatives.

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9

Igudesman, Konstantin, Marsel Davletbaev, and Gleb Shabernev. "New Approach to Fractal Approximation of Vector-Functions." Abstract and Applied Analysis 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/278313.

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This paper introduces new approach to approximation of continuous vector-functions and vector sequences by fractal interpolation vector-functions which are multidimensional generalization of fractal interpolation functions. Best values of fractal interpolation vector-functions parameters are found. We give schemes of approximation of some sets of data and consider examples of approximation of smooth curves with different conditions.

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10

NAVASCUÉS, M. A., S. K. KATIYAR, and A. K. B. CHAND. "MULTIVARIATE AFFINE FRACTAL INTERPOLATION." Fractals 28, no. 07 (November 2020): 2050136. http://dx.doi.org/10.1142/s0218348x20501364.

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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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11

LIANG, YONG-SHUN, and QI ZHANG. "A TYPE OF FRACTAL INTERPOLATION FUNCTIONS AND THEIR FRACTIONAL CALCULUS." Fractals 24, no. 02 (June 2016): 1650026. http://dx.doi.org/10.1142/s0218348x16500262.

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Combine Chebyshev systems with fractal interpolation, certain continuous functions have been approximated by fractal interpolation functions unanimously. Local structure of these fractal interpolation functions (FIF) has been discussed. The relationship between order of Riemann–Liouville fractional calculus and Box dimension of FIF has been investigated.

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12

Feng, Zhigang, and Xiuqing Sun. "Box-counting dimensions of fractal interpolation surfaces derived from fractal interpolation functions." Journal of Mathematical Analysis and Applications 412, no. 1 (April 2014): 416–25. http://dx.doi.org/10.1016/j.jmaa.2013.10.032.

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13

Vijender, Nallapu, and Vasileios Drakopoulos. "On the Bernstein Affine Fractal Interpolation Curved Lines and Surfaces." Axioms 9, no. 4 (October 18, 2020): 119. http://dx.doi.org/10.3390/axioms9040119.

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In this article, firstly, an overview of affine fractal interpolation functions using a suitable iterated function system is presented and, secondly, the construction of Bernstein affine fractal interpolation functions in two and three dimensions is introduced. Moreover, the convergence of the proposed Bernstein affine fractal interpolation functions towards the data generating function does not require any condition on the scaling factors. Consequently, the proposed Bernstein affine fractal interpolation functions possess irregularity at any stage of convergence towards the data generating function.

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14

Massopust, Peter R. "Local Fractal Interpolation on Unbounded Domains." Proceedings of the Edinburgh Mathematical Society 61, no. 1 (January 23, 2018): 151–67. http://dx.doi.org/10.1017/s0013091517000268.

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AbstractWe define fractal interpolation on unbounded domains for a certain class of topological spaces and construct local fractal functions. In addition, we derive some properties of these local fractal functions, consider their tensor products, and give conditions for local fractal functions on unbounded domains to be elements of Bochner–Lebesgue spaces.

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15

Ri, Song-Il, Vasileios Drakopoulos, and Song-Min Nam. "Fractal Interpolation Using Harmonic Functions on the Koch Curve." Fractal and Fractional 5, no. 2 (April 5, 2021): 28. http://dx.doi.org/10.3390/fractalfract5020028.

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The Koch curve was first described by the Swedish mathematician Helge von Koch in 1904 as an example of a continuous but nowhere differentiable curve. Such functions are now characterised as fractal since their graphs are in general fractal sets. Furthermore, it can be obtained as the graph of an appropriately chosen iterated function system. On the other hand, a fractal interpolation function can be seen as a special case of an iterated function system thus maintaining all of its characteristics. Fractal interpolation functions are continuous functions that can be used to model continuous signals. An in-depth discussion on the theory of affine fractal interpolation functions generating the Koch Curve by using fractal analysis as well as its recent development including some of the research made by the authors is provided. We ensure that the graph of fractal interpolation functions on the Koch Curve are attractors of an iterated function system constructed by non-constant harmonic functions.

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16

Navascués, María Antonia, Arya Kumar Bedabrata Chand, Viswanathan Puthan Veedu, and María Victoria Sebastián. "Fractal Interpolation Functions: A Short Survey." Applied Mathematics 05, no. 12 (2014): 1834–41. http://dx.doi.org/10.4236/am.2014.512176.

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17

DALLA, LEONI. "BIVARIATE FRACTAL INTERPOLATION FUNCTIONS ON GRIDS." Fractals 10, no. 01 (March 2002): 53–58. http://dx.doi.org/10.1142/s0218348x02000951.

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18

Chand, A. K. B., and G. P. Kapoor. "Generalized Cubic Spline Fractal Interpolation Functions." SIAM Journal on Numerical Analysis 44, no. 2 (January 2006): 655–76. http://dx.doi.org/10.1137/040611070.

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19

Barnsley, Michael F., and Andrew N. Harrington. "The calculus of fractal interpolation functions." Journal of Approximation Theory 57, no. 1 (April 1989): 14–34. http://dx.doi.org/10.1016/0021-9045(89)90080-4.

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20

Chand, A. K. B., and N. Vijender. "Monotonicity Preserving Rational Quadratic Fractal Interpolation Functions." Advances in Numerical Analysis 2014 (January 30, 2014): 1–17. http://dx.doi.org/10.1155/2014/504825.

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Fractal interpolation is an advanced technique for analysis and synthesis of scientific and engineering data. We introduce the 𝒞1-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 𝒞1-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 𝒞4. 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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21

YUN, CHOL-HUI. "HIDDEN VARIABLE RECURRENT FRACTAL INTERPOLATION FUNCTIONS WITH FUNCTION CONTRACTIVITY FACTORS." Fractals 27, no. 07 (November 2019): 1950113. http://dx.doi.org/10.1142/s0218348x19501135.

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In this paper, we introduce a construction of hidden variable recurrent fractal interpolation functions (HVRFIFs) with four function contractivity factors. The HVRFIF is a hidden variable fractal interpolation function (HVFIF) constructed using a recurrent iterated function system (RIFS). In the fractal interpolation theory, it is very important to ensure flexibility and diversity of the construction of interpolation functions. RIFSs produce fractal sets with local self-similarity structure. Therefore, the RIFS can describe the irregular and complicated objects in nature better than the iterated function system (IFS). The HVFIF is neither self-similar nor self-affine one. Hence, the HVFIF is more complicated, diverse and irregular than the fractal interpolation function (FIF). The contractivity factors of IFS are very important one that determines characteristics of FIFs. The IFS and RIFS with function contractivity factors can describe the fractal objects in nature better than one with constant contractivity factors. To ensure higher flexibility and diversity of the construction of the FIFs, we present constructions of one variable HVRFIFs and bivariable HVRFIFs using RIFS with four function contractivity factors.

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22

KIM, JINMYONG, HYONJIN KIM, and HAKMYONG MUN. "CONSTRUCTION OF NONLINEAR HIDDEN VARIABLE FRACTAL INTERPOLATION FUNCTIONS AND THEIR STABILITY." Fractals 27, no. 06 (September 2019): 1950103. http://dx.doi.org/10.1142/s0218348x19501032.

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This paper presents a method to construct nonlinear hidden variable fractal interpolation functions (FIFs) and their stability results. We ensure that the projections of attractors of vector-valued nonlinear iterated function systems (IFSs) constructed by Rakotch contractions and function vertical scaling factors are graphs of some continuous functions interpolating the given data. We also give an explicit example illustrating obtained results. Then, we get the stability results of the constructed FIFs in the case of the generalized interpolation data having small perturbations.

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23

NAVASCUÉS, M. A., P. VISWANATHAN, A. K. B. CHAND, M. V. SEBASTIÁN, and S. K. KATIYAR. "FRACTAL BASES FOR BANACH SPACES OF SMOOTH FUNCTIONS." Bulletin of the Australian Mathematical Society 92, no. 3 (July 30, 2015): 405–19. http://dx.doi.org/10.1017/s0004972715000738.

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This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley–Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to the present setting. The general result is applied on a special class of iterated function systems in order to develop differentiability of the so-called $\boldsymbol{{\it\alpha}}$-fractal functions. This leads to a bounded linear map on the space ${\mathcal{C}}^{k}(I)$ which is exploited to prove the existence of a Schauder basis for ${\mathcal{C}}^{k}(I)$ consisting of smooth fractal functions.

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24

Xie, Heping, and Hongquan Sun. "The Study on Bivariate Fractal Interpolation Functions and Creation of Fractal Interpolated Surfaces." Fractals 05, no. 04 (December 1997): 625–34. http://dx.doi.org/10.1142/s0218348x97000504.

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In this paper, the methods of construction of a fractal surface are introduced, the principle of bivariate fractal interpolation functions is discussed. The theorem of the uniqueness of an iterated function system of bivariate fractal interpolation functions is proved. Moreover, the theorem of fractal dimension of fractal interpolated surface is derived. Based on these theorems, the fractal interpolated surfaces are created by using practical data.

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25

Chand, A. K. B., and G. P. Kapoor. "Spline coalescence hidden variable fractal interpolation functions." Journal of Applied Mathematics 2006 (2006): 1–17. http://dx.doi.org/10.1155/jam/2006/36829.

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This paper generalizes the classical spline using a new construction of spline coalescence hidden variable fractal interpolation function (CHFIF). The derivative of a spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of a nondiagonal iterated function system. Our construction generalizes the construction of Barnsley and Harrington (1989), when the construction is not restricted to a particular type of boundary conditions. Spline CHFIFs are likely to be potentially useful in approximation theory due to effects of the hidden variables and these effects are demonstrated through suitable examples in the present work.

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26

Massopust, Peter. "Non-Stationary Fractal Interpolation." Mathematics 7, no. 8 (July 25, 2019): 666. http://dx.doi.org/10.3390/math7080666.

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We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps { F k } k ∈ N where each F k maps H ( X ) → H ( X ) and arises from an iterated function system. Employing the recently-developed theory of non-stationary versions of fixed points and the concept of forward and backward trajectories, we present new classes of fractal functions exhibiting different local and global behavior and extend fractal interpolation to this new, more flexible setting.

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27

BOUBOULIS, P., and L. DALLA. "HIDDEN VARIABLE VECTOR VALUED FRACTAL INTERPOLATION FUNCTIONS." Fractals 13, no. 03 (September 2005): 227–32. http://dx.doi.org/10.1142/s0218348x05002854.

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28

Somogyi, Ildiko, and Anna Soos. "Graph-directed random fractal interpolation function." Studia Universitatis Babes-Bolyai Matematica 66, no. 2 (June 15, 2021): 247–55. http://dx.doi.org/10.24193/subbmath.2021.2.01.

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"Barnsley introduced in [1] the notion of fractal interpolation function (FIF). He said that a fractal function is a (FIF) if it possess some interpolation properties. It has the advantage that it can be also combined with the classical methods or real data interpolation. Hutchinson and Ruschendorf [7] gave the stochastic version of fractal interpolation function. In order to obtain fractal interpolation functions with more exibility, Wang and Yu [9] used instead of a constant scaling parameter a variable vertical scaling factor. Also the notion of fractal interpolation can be generalized to the graph-directed case introduced by Deniz and Ozdemir in [5]. In this paper we study the case of a stochastic fractal interpolation function with graph-directed fractal function."

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29

Price, J. R., and M. H. Hayes. "Resampling and reconstruction with fractal interpolation functions." IEEE Signal Processing Letters 5, no. 9 (September 1998): 228–30. http://dx.doi.org/10.1109/97.712106.

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30

Luor, Dah-Chin. "Fractal interpolation functions with partial self similarity." Journal of Mathematical Analysis and Applications 464, no. 1 (August 2018): 911–23. http://dx.doi.org/10.1016/j.jmaa.2018.04.041.

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31

Li, Xiao-hui, and Huo-jun Ruan. "Energy and Laplacian of fractal interpolation functions." Applied Mathematics-A Journal of Chinese Universities 32, no. 2 (June 2017): 201–10. http://dx.doi.org/10.1007/s11766-017-3482-8.

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32

Navascués, M. A., and M. V. Sebastián. "Generalization of Hermite functions by fractal interpolation." Journal of Approximation Theory 131, no. 1 (November 2004): 19–29. http://dx.doi.org/10.1016/j.jat.2004.09.001.

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33

Luor, Dah-Chin. "Fractal interpolation functions for random data sets." Chaos, Solitons & Fractals 114 (September 2018): 256–63. http://dx.doi.org/10.1016/j.chaos.2018.06.033.

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34

CHAND, A. K. B., P. VISWANATHAN, and K. M. REDDY. "TOWARDS A MORE GENERAL TYPE OF UNIVARIATE CONSTRAINED INTERPOLATION WITH FRACTAL SPLINES." Fractals 23, no. 04 (December 2015): 1550040. http://dx.doi.org/10.1142/s0218348x15500401.

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Recently, in [Electron. Trans. Numer. Anal. 41 (2014) 420–442] authors introduced a new class of rational cubic fractal interpolation functions with linear denominators via fractal perturbation of traditional nonrecursive rational cubic splines and investigated their basic shape preserving properties. The main goal of the current paper is to embark on univariate constrained fractal interpolation that is more general than what was considered so far. To this end, we propose some strategies for selecting the parameters of the rational fractal spline so that the interpolating curves lie strictly above or below a prescribed linear or a quadratic spline function. Approximation property of the proposed rational cubic fractal spine is broached by using the Peano kernel theorem as an interlude. The paper also provides an illustration of background theory, veined by examples.

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35

NAVASCUÉS, M. ANTONIA, and M. VICTORIA SEBASTIÁN. "SOME RESULTS OF CONVERGENCE OF CUBIC SPLINE FRACTAL INTERPOLATION FUNCTIONS." Fractals 11, no. 01 (March 2003): 1–7. http://dx.doi.org/10.1142/s0218348x03001550.

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Fractal interpolation functions (FIFs) provide new methods of approximation of experimental data. In the present paper, a fractal technique generalizing cubic spline functions is proposed. A FIF f is defined as the fixed point of a map between spaces of functions. The properties of this correspondence allow to deduce some inequalities that express the sensitivity of these functions and their derivatives to those changes in the parameters defining them. Under some hypotheses on the original function, bounds of the interpolation error for f, f′ and f′′ are obtained. As a consequence, the uniform convergence to the original function and its derivative as the interpolation step tends to zero is proved. According to these results, it is possible to approximate, with arbitrary accuracy, a smooth function and its derivatives by using a cubic spline fractal interpolation function (SFIF).

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36

Li, Jing, and Weiyi Su. "The Smoothness of Fractal Interpolation Functions onℝand onp-Series Local Fields." Discrete Dynamics in Nature and Society 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/904576.

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A fractal interpolation function on ap-series local fieldKpis defined, and itsp-type smoothness is shown by virtue of the equivalent relationship between the Hölder type spaceCσKpand the Lipschitz class Lipσ,Kp. The orders of thep-type derivatives and the fractal dimensions of the graphs of Weierstrass type function on local fields are given as an example. Theα-fractal function onℝis introduced and the conclusion of its smoothness is improved in a more general case; some examples are shown to support the conclusion. Finally, a comparison between the fractal interpolation functions defined onℝandKpis given.

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37

VERMA, S., and P. VISWANATHAN. "A REVISIT TO α-FRACTAL FUNCTION AND BOX DIMENSION OF ITS GRAPH." Fractals 27, no. 06 (September 2019): 1950090. http://dx.doi.org/10.1142/s0218348x19500907.

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One of the tools offered by fractal geometry is fractal interpolation, which forms a basis for the constructive approximation theory for nondifferentiable functions. The notion of fractal interpolation function can be used to obtain a wide spectrum of self-referential functions associated to a prescribed continuous function on a compact interval in [Formula: see text]. These fractal maps, the so-called [Formula: see text]-fractal functions, are defined by means of suitable iterated function system which involves some parameters. Building on the literature related to the notion of [Formula: see text]-fractal functions, the current study targets to record the continuous dependence of the [Formula: see text]-fractal function on parameters involved in its definition. Furthermore, the paper attempts to study the box dimension of the graph of the [Formula: see text]-fractal function.

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CHAND, A. K. B., and K. R. TYADA. "PARTIALLY BLENDED CONSTRAINED RATIONAL CUBIC TRIGONOMETRIC FRACTAL INTERPOLATION SURFACES." Fractals 24, no. 03 (August 30, 2016): 1650027. http://dx.doi.org/10.1142/s0218348x16500274.

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Fractal interpolation is an advance technique for visualization of scientific shaped data. In this paper, we present a new family of partially blended rational cubic trigonometric fractal interpolation surfaces (RCTFISs) with a combination of blending functions and univariate rational trigonometric fractal interpolation functions (FIFs) along the grid lines of the interpolation domain. The developed FIFs use rational trigonometric functions [Formula: see text], where [Formula: see text] and [Formula: see text] are cubic trigonometric polynomials with four shape parameters. The convergence analysis of partially blended RCTFIS with the original surface data generating function is discussed. We derive sufficient data-dependent conditions on the scaling factors and shape parameters such that the fractal grid line functions lie above the grid lines of a plane [Formula: see text], and consequently the proposed partially blended RCTFIS lies above the plane [Formula: see text]. Positivity preserving partially blended RCTFIS is a special case of the constrained partially blended RCTFIS. Numerical examples are provided to support the proposed theoretical results.

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KATIYAR, S. K., and A. K. B. CHAND. "SHAPE PRESERVING RATIONAL QUARTIC FRACTAL FUNCTIONS." Fractals 27, no. 08 (December 2019): 1950141. http://dx.doi.org/10.1142/s0218348x1950141x.

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The appearance of fractal interpolation function represents a revival of experimental mathematics, raised by computers and intensified by powerful evidence of its applications. This paper is devoted to establish a method to construct [Formula: see text]-fractal rational quartic spline, which eventually provides a unified approach for the generalization of various traditional nonrecursive rational splines involving shape parameters. We deduce the uniform error bound for the [Formula: see text]-fractal rational quartic spline when the original function is in [Formula: see text]. By solving a system of linear equations, appropriate values of the derivative parameters are determined so as to enhance the continuity of the [Formula: see text]-fractal rational quartic spline to [Formula: see text]. The elements of the iterated function system are identified befittingly so that the class of [Formula: see text]-fractal function [Formula: see text] incorporates the geometric features such as positivity, monotonicity and convexity in addition to the regularity inherent in the germ [Formula: see text]. This general theory in conjunction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions.

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40

WANG, HONG-YONG, and JIA-BING JI. "SURFACE FITTING AND ERROR ANALYSIS USING FRACTAL INTERPOLATION." International Journal of Bifurcation and Chaos 22, no. 08 (August 2012): 1250194. http://dx.doi.org/10.1142/s0218127412501945.

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The fitting of a given continuous surface defined on a rectangular region in ℝ2 is studied by using a fractal interpolation surface, and the error analysis of fitting is made in this paper. The fractal interpolation functions used in surface fitting are generated by a special class of iterated function systems. Some properties of such fractal interpolation functions are discussed. Moreover, the error problems of fitting are investigated by using an operator defined on the space of continuous functions, and the upper estimates of errors are obtained in the sense of two kinds of metrics. Finally, a specific numerical example to illustrate the application of the procedure is also described.

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41

LUOR, DAH-CHIN. "AUTOCOVARIANCE AND INCREMENTS OF DEVIATION OF FRACTAL INTERPOLATION FUNCTIONS FOR RANDOM DATASETS." Fractals 26, no. 05 (October 2018): 1850075. http://dx.doi.org/10.1142/s0218348x18500755.

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In this paper we consider the expectation, the autocovariance, and increments of the deviation of a fractal interpolation function [Formula: see text] corresponding to a random dataset [Formula: see text]. We show that the covariance of [Formula: see text] and [Formula: see text] is a fractal interpolation function on [Formula: see text] for each fixed [Formula: see text], where [Formula: see text]. We also prove that, for a fixed [Formula: see text], the covariance of [Formula: see text] and [Formula: see text] is a fractal interpolation function on [Formula: see text]. A special type of increments of the deviation of [Formula: see text] is also investigated.

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42

Sneha and K. Katiyar. "CONSTRAINED SHAPE PRESERVING RATIONAL QUINTIC FRACTAL INTERPOLATION FUNCTIONS." Advances in Mathematics: Scientific Journal 9, no. 8 (August 15, 2020): 5521–35. http://dx.doi.org/10.37418/amsj.9.8.22.

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43

Chand, A. K. B., and K. R. Tyada. "Constrained shape preserving rational cubic fractal interpolation functions." Rocky Mountain Journal of Mathematics 48, no. 1 (February 2018): 75–105. http://dx.doi.org/10.1216/rmj-2018-48-1-75.

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Akhtar, Md Nasim, and M. Guru Prem Prasad. "Graph-Directed Coalescence Hidden Variable Fractal Interpolation Functions." Applied Mathematics 07, no. 04 (2016): 335–45. http://dx.doi.org/10.4236/am.2016.74031.

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FENG, ZHIGANG, LIXIN TIAN, and JIANLI JIAO. "INTEGRATION AND FOURIER TRANSFORM OF FRACTAL INTERPOLATION FUNCTIONS." Fractals 13, no. 01 (March 2005): 33–41. http://dx.doi.org/10.1142/s0218348x05002726.

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Abstract:

Fractal interpolation function (FIF) is continuous on its interval of definition. As a special kind of continuous function, FIFs' integrations on various scales and Fourier transform are studied in this paper. All of them can be expressed by the parameters of the corresponding iterative function systems.

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Donovan, George C., Jeffrey S. Geronimo, Douglas P. Hardin, and Peter R. Massopust. "Construction of Orthogonal Wavelets Using Fractal Interpolation Functions." SIAM Journal on Mathematical Analysis 27, no. 4 (July 1996): 1158–92. http://dx.doi.org/10.1137/s0036141093256526.

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Gang, Chen. "The smoothness and dimension of fractal interpolation functions." Applied Mathematics 11, no. 4 (December 1996): 409–18. http://dx.doi.org/10.1007/bf02662880.

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Li, J., P. Yip, and E. Boss�. "Fractal interpolation functions and data compression for tracking." Circuits Systems and Signal Processing 16, no. 1 (January 1997): 59–67. http://dx.doi.org/10.1007/bf01183175.

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Serpa, Cristina, and Jorge Buescu. "Explicitly defined fractal interpolation functions with variable parameters." Chaos, Solitons & Fractals 75 (June 2015): 76–83. http://dx.doi.org/10.1016/j.chaos.2015.01.023.

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Tyada, K. R., A. K. B. Chand, and M. Sajid. "Shape preserving rational cubic trigonometric fractal interpolation functions." Mathematics and Computers in Simulation 190 (December 2021): 866–91. http://dx.doi.org/10.1016/j.matcom.2021.06.015.

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