In this paper, we continue a study of approximation properties of the fuzzy transform (F-transform) with the partition generated by the Shepard kernel. We make the error estimate in terms of a modulus of continuity which has a higher order than the previously known. Also, we obtain the error estimate of the iterative F-transform-based method for linear Fredholm integral equations of the second kind.
BedeB. and RudasI.J., Approximation properties of fuzzy transforms, Fuzzy Sets and Systems180 (2011), 20–40.
2.
BaghmishehM. and EzzatiR., Error estimation and numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using triangular functions, Intelligent and Fuzzy Systems30(2) (2016), 639–649.
3.
BaghmishehM. and EzzatiR., Numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using hybrid of black–pulse functions and Taylor series, Advances in Difference Equations51 (2015), 1–15. DOI: 10.1186/s13662-015-0389-7
4.
CollinsP.J., Differential and Integral Equations, Oxford University Press, 2006.
5.
DaňkováM. and
ŠtěpničkaM., Fuzzy transform as an additive normal form, Fuzzy Sets and Systems157(8) (2006), 1024–1035.
6.
DelvesL.M. and MohamedJ.L., Computational methods for integral equation, Cambridge University Press, 1985.
7.
MalinowskiM.T., Random fuzzy fractional integral equations – theoretical foundations, Fuzzy Sets and Systems265 (2015), 39–62.
8.
Di MartinoF.,
HurtikP.,
PerfilievaI. and SessaS., A color image reduction based on Fuzzy transforms, Information Sciences266 (2014), 101–114.
9.
Di MartinoF.,
LoiaV. and SessaS., Fuzzy transforms method and attribute dependency in data analysis, Information Sciences180 (2010), 493–505.
10.
DiMartinoF., LoiaV. and SessaS., Fuzzy transforms for compression and decompression of color videos, Information Sciences180(20) (2010), 3914–3931.
11.
Di MartinoF.,
LoiaV.,
PerfilievaI. and SessaS., An image coding/decoding method based on direct and inverse fuzzy transforms, International Journal of Approximate Reasoning48 (2008), 110–131.
12.
Di MartinoF. and
SessaS., Compression and decompression of images with discrete fuzzy transforms, Information Sciences177 (2007), 2349–2362.
13.
KharabA. and GuentherR.B., An Introduction To Numerical Methods: A MATLAB Approach, Chapman and Hall/CRC, 2011.
14.
PerfilievaI., HurtikP., Di MartinoF. and
SessaS., Image reduction method based on the F-transform, Soft Computing (2015). DOI: 10.1007/s00500-015-1885-0
15.
KhastanA., PerfilievaI. and AlijaniZ., A new fuzzyapproximation method to Cauchy problems by fuzzy transform, Fuzzy Sets and Systems288 (2016), 75–95.
16.
PerfilievaI., DaňkováM.M. and
BedeB., Towards a higher degree F-transform, Fuzzy Sets and Systems180 (2011), 3–19.
17.
PerfilievaI. and De BaetsB., Fuzzy transforms of monotone functions with application to image, Information Sciences180(17) (2010), 3304–3315.
18.
PerfilievaI., NovàkV. and DvoràkA., Fuzzy transforms in the analysis of data, International Journal of Approximate Reasoning48 (2008), 36–46.
19.
PerfilievaI., Fuzzy transforms: Theory and applications, Fuzzy Sets and Systems157 (2006), 9930–1023.
20.
PerfilievaI., Fuzzy transforms, in:
PetersJ.F., et al. (Eds.), Transaction s on Rough Sets II, Lecture Notes in Computer Science, vol. 3135, 2004, pp. 63–81.
21.
ŠtěpničkaM. and
PolakovičO., A neural network approach to the fuzzy transform, Fuzzy Sets an Systems160 (2009), 1037–1047.
22.
XiaoW. and ZhouS.P., A Jackson type estimate for Shepard operators in Lp spaces for p > 1, Acta Mathematica Hungarica95 (2002), 217–224.
23.
ZiariS. and BicaA.M., New error estimate in the iterative numerical method for nonlinear fuzzy Hammerstein-Fredholm integral equations, Fuzzy Sets and Systems295 (2016), 136–152.