Fuzzy projection versus inverse fuzzy transform as sampling/interpolation schemes

Adjoint Fuzzy Partition and Generalized Sampling Theorem

“…In Takagi-Sugeno models, the coefficients guarantee that the corresponding approximating function is a best approximation on a whole domain in the sense of the L 2 metric. Similar models have been considered in [1,7].…”

“…In [8], it has been shown that the inverse F-transform can approximate a continuous function with an arbitrary precision. In the later publications [1,7], other smooth approximations for functions from L 2 (R) by the inverse F-transforms were proposed.…”

Non-additive interval-valued F-transform

“…As pointed out by Perfilieva and most of the authors [23,21,3,1,32], a function cannot be reconstructed by using its F-transform, since transforming consists of sampling, and sampling leads to information loss. However, the reconstructed function can approximate its original continuous function with an arbitrary precision.…”

“…Thus, in this setting, another basis should be preferred that leads to minimizing a global L 2 distance. As shown in [3], this leads to the least square interpolation procedure conventionally used in signal processing. However, this approach only applies to discrete functions.…”

Noise Reduction on Numerical Solutions of Partial Differential Equations using Fuzzy Transform

“…The inverse F-transform properties of sampling and interpolation can be improved by using the so-called fuzzy projection [1]. The ideas of fuzzy projection will not be used in this study because it does not permit to created a transform methodology to apply on numerical solutions of PDEs.…”