Construction of new fractal interpolation functions through integration method. (English) Zbl 1501.28005
Summary: This paper investigates the classical integral of various types of fractal interpolation functions namely linear fractal interpolation function, \(\alpha\)-fractal function and hidden variable fractal interpolation function with function scaling factors. The integral of a fractal function is again a fractal function to a different set of interpolation data if the integral of fractal function is predefined at the initial point or end point of the given data. In this study, the selection of vertical scaling factors as continuous functions on the closed interval of \(\mathbb{R}\) provides more diverse fractal interpolation functions compared to the fractal interpolations functions with constant scaling factors.
MSC:
| 28A80 | Fractals |
| 41A05 | Interpolation in approximation theory |
| 97I50 | Integral calculus (educational aspects) |
References:
| [1] | Barnsley, MF, Fractal functions and interpolation, Constr. Approx., 2, 1, 303-329 (1986) · Zbl 0606.41005 · doi:10.1007/BF01893434 |
| [2] | Barnsley, MF, Fractals Everywhere (1993), New York: Academic Press, New York · Zbl 0691.58001 |
| [3] | Banerjee, S.; Easwaramoorthy, D.; Gowrisankar, A., Fractal Functions, Dimensions and Signal Analysis (2011), Cham: Springer, Cham · Zbl 1458.94072 |
| [4] | Pacurar, CM, A countable fractal interpolation scheme involving Rakotch contractions, Results Math., 76, 161 (2021) · Zbl 1472.28011 · doi:10.1007/s00025-021-01470-x |
| [5] | Banerjee, S.; Hassan, MK; Mukherjee, S.; Gowrisankar, A., Fractal Patterns in Nonlinear Dynamics and Applications (2020), Baco Raton: CRC Press, Baco Raton · Zbl 1445.28001 · doi:10.1201/9781315151564 |
| [6] | Barnsley, MF; Harrington, AN, The calculus of fractal interpolation functions, J. Approx. Theory, 57, 14-34 (1989) · Zbl 0693.41008 · doi:10.1016/0021-9045(89)90080-4 |
| [7] | Peng, WL; Yao, K.; Zhang, X.; Yao, J., Box dimension of Weyl-Marchaud fractional derivative of linear fractal interpolation functions, Fractals, 27, 4, 1950058 (2019) · Zbl 1433.28025 · doi:10.1142/S0218348X19500580 |
| [8] | Priyanka, TMC; Gowrisankar, A., Analysis on Wely-Marchaud fractional derivative of types of fractal interpolation function with fractal dimension, Fractals, 29, 7, 2150215 (2021) · Zbl 1511.28007 · doi:10.1142/S0218348X21502157 |
| [9] | Kui, YW; Yao, XZ, On the Hadamard fractional calculus of a fractal function, Fractals, 26, 3, 1850025 (2018) · Zbl 1433.26009 · doi:10.1142/S0218348X18500251 |
| [10] | Verma, S.; Viswanathan, P., Katugampola fractional integral and fractal dimension of bivariate functions, Results Math., 76, 165 (2021) · Zbl 1477.28009 · doi:10.1007/s00025-021-01475-6 |
| [11] | Verma, S.; Viswanathan, P., A note on Katugampola fractional calculus and fractal dimensions, Appl. Math. Comput., 339, 220-230 (2018) · Zbl 1428.28018 |
| [12] | Ri, M-G; Yun, C-H, Riemann Liouville fractional integral of hidden variable fractal interpolation function, Chaos Solitons Fract., 140, 110126 (2020) · Zbl 1495.26012 · doi:10.1016/j.chaos.2020.110126 |
| [13] | Gowrisankar, A.; Guru Prem Prasad, M., Riemann-Liouville calculus on quadratic fractal interpolation function with variable scaling factors, J. Anal., 27, 2, 347-363 (2019) · Zbl 1423.28021 · doi:10.1007/s41478-018-0133-2 |
| [14] | Wang, H-Y; Jia-Shan, Yu, Fractal interpolation functions with variable parameters and their analytical properties, J. Approx. Theory, 175, 1-18 (2013) · Zbl 1303.28014 · doi:10.1016/j.jat.2013.07.008 |
| [15] | Barnsley, MF; Elton, J.; Hardin, D.; Massopust, P., Hidden variable fractal interpolation functions, SIAM J. Math. Anal., 20, 5, 1218-1242 (1989) · Zbl 0704.26009 · doi:10.1137/0520080 |
| [16] | Yun, C-H, Hidden variable recurrent fractal interpolation functions with function contractivity factors, Fractals, 27, 7, 1950113 (2019) · Zbl 1434.28042 · doi:10.1142/S0218348X19501135 |
| [17] | Vijender, N., Approximation by hidden variable fractal functions: a sequential approach, Results Math., 74, 192 (2019) · Zbl 1428.28019 · doi:10.1007/s00025-019-1114-8 |
| [18] | Navascués, MA, Fractal polynomial interpolation, Z. Anal. Anwend., 25, 2, 401-418 (2005) · Zbl 1082.28006 · doi:10.4171/ZAA/1248 |
| [19] | Navascués, MA, Non-smooth polynomial., Int. J. Math. Anal., 1, 4, 159-174 (2007) · Zbl 1134.28302 |
| [20] | Nasim Akhtara, Md; Guru Prem Prasad, M.; Navascués, MA, Box dimension of \(alpha\)-fractal function with variable scaling factors in subintervals, Chaos Solitons Fract., 103, 440-449 (2017) · Zbl 1375.28011 · doi:10.1016/j.chaos.2017.07.002 |
| [21] | Chand, AKB; Kapoor, GP, Spline coalescence hidden variable fractal interpolation function, J. Appl. Math., 1-17, 36829 (2006) · Zbl 1146.41003 |
| [22] | Kapoor, GP; Prasad, SA, Cubic spline super fractal interpolation functions, Fractals, 22, 1-2, 1450005 (2014) · Zbl 1306.28009 · doi:10.1142/S0218348X14500054 |
| [23] | Kapoor, GP; Chand, AKB, Generalized cubic spline fractal interpolation functions, SIAM J. Numer. Anal., 44, 655-676 (2006) · Zbl 1136.41006 · doi:10.1137/040611070 |
| [24] | Priyanka, TMC; Gowrisankar, A., Riemann-Liouville fractional integral of non-affine fractal interpolation function and its fractional operator, Eur. Phys. J. Spec. Top., 230, 3789-3805 (2021) · doi:10.1140/epjs/s11734-021-00315-6 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
