On the decomposition of functions as sum and product in terms of various fractal dimensions. (English) Zbl 07928821
Summary: If local forms of Littlewood’s three principles are stated as axioms for an ordered field, then each principle is equivalent to the completeness axiom.
References:
| [1] | V. Agrawal and T. Som, Fractal dimension of \(\alpha \)-fractal function on the Sierpiński gasket, Eur. Phys. J. Spec. Top., 230 (2021), 3781-3787. |
| [2] | V. Agrawal and T. Som, \( \mathcal{L}^p\)-approximation using fractal functions on the Sierpiński gasket, Results Math., 77(2) (2022), Art. No. 74. · Zbl 1486.28005 |
| [3] | F. Bayart and Y. Heurteaux, On the Hausdorff dimension of graphs of prevalent continuous functions on compact sets. Further Developments in Fractals and Related Fields, 25-34. Trends in Mathematics, Birkhäuser, New York, 2013. · Zbl 1268.28003 |
| [4] | S. Chandra and S. Abbas, Analysis of fractal dimension of mixed Riemann-Liouville integral, Numer. Algorithms, 91(3) (2022), 1021-1046. · Zbl 1515.28008 |
| [5] | S. Chandra and S. Abbas, On fractal dimensions of fractal functions using function spaces, Bull. Austral. Math. Soc., 106(3) (2022), 470-480. · Zbl 1528.28016 |
| [6] | S. Chandra, S. Abbas, and S. Verma, Bernstein super fractal interpolation function for countable data systems, Numer. Algorithms, 92(4) (2023), 2457-2481. · Zbl 1529.28009 |
| [7] | K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Third edition. Wiley, Chichester, 2014. · Zbl 1285.28011 |
| [8] | K. J. Falconer and J. M. Fraser, The horizon problem for prevalent surfaces, Math. Proc. Cambridge Philos. Soc., 151(2) (2011), 355-372. · Zbl 1235.28005 |
| [9] | K. J. Falconer and J. D. Howroyd, Projection theorems for box and packing dimensions, Math. Proc. Cambridge Philos. Soc., 119(2) (1996), 287-295. · Zbl 0846.28004 |
| [10] | J. M. Fraser, Assouad Dimension and Fractal Geometry, Cambridge Tracts in Mathematics, 222, Cambridge University Press, Cambridge, 2021. · Zbl 1467.28001 |
| [11] | P. D. Humke and G. Petruska, The packing dimension of a typical continuous function is \(2\), Real Anal. Exchange, 14(2) (1988/89), 345-358. · Zbl 0678.26002 |
| [12] | J. Hyde, V. Laschos, L. Olsen, I. Petrykiewicz, and A. Shaw, On the box dimensions of graphs of typical continuous functions, J. Math. Anal. Appl., 391(2) (2012), 567-581. · Zbl 1238.28005 |
| [13] | S. Jha and S. Verma, Dimensional analysis of \(\alpha \)-fractal functions, Results Math., 76(4) (2021), Art. No. 186. · Zbl 1478.28009 |
| [14] | Y. S. Liang, Box dimensions of Riemann-Liouville fractional integrals of continuous functions of bounded variation, onlinear Anal., 72(11) (2010), 4304-4306. · Zbl 1190.28001 |
| [15] | J. Liu and D. Liu, On the decomposition of continuous functions and dimensions, Fractals, 28(1) (2020), Art. No. 2050007. · Zbl 1434.28007 |
| [16] | J. Liu, B. Tan, and J. Wu, Graphs of continuous functions and packing dimension, J. Math. Anal. Appl., 435(2) (2016), 1099-1106. · Zbl 1329.28012 |
| [17] | J. Liu and J. Wu, A remark on decomposition of continuous functions, J. Math. Anal. Appl., 401(1) (2013), 404-406. · Zbl 1261.28008 |
| [18] | R. D. Mauldin and S. C. Williams, On the Hausdorff dimension of some graphs, Trans. Amer. Math. Soc., 298(2) (1986), 793-803. · Zbl 0603.28003 |
| [19] | S. A. Prasad and S. Verma, Fractal interpolation function on products of the Sierpiński gaskets, Chaos Solitons Fractals, 166 (2023), Art. No. 112988. |
| [20] | M. Verma and A. Priyadarshi, Graphs of continuous functions and fractal dimensions, Chaos Solitons Fractals, 172 (2023), Art. No. 113513. |
| [21] | P. Wingren, Dimensions of graphs of functions and lacunary decompositions of spline approximations, Real Anal. Exchange, 26(1) (2000/01), 17-26. · Zbl 1016.28007 |
| [22] | B. Y. Yu and Y. S. Liang, Fractal dimension variation of continuous functions under certain operations, Fractals, 31(5) (2023), Art. No. 2350044. · Zbl 1532.28018 |
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.
