The goal of the paper is to enlarge the class of fuzzy transforms (F-transforms) by considering different types of fuzzy partitions. \ New types of F-transforms are constructed based on B-splines, Shepard kernels, Bernstein basis polynomials and Favard-Szaz-Mirakjan type operators. The approximation properties of these F-transforms are studied. A fuzzy partition of an interval \(I=\left[ a,b\right] \) is defined as a sequence \({\mathbf A}=\{A_{1},A_{2},\dots,A_{k}\}\) of fuzzy sets (called atoms of the partition) \(A_{i}:I\rightarrow[0,1],\;i=1,\dots,k\) such that \(\sum_{i=1}^{k}A_{i}(x)=1\) for all \(x\in I\). Let \(a=y_{1}0\right\} \subseteq[ y_{i},y_{i+r}\}\), then \(A\) is said to be a fuzzy partition with small support. Let \(f\) be a continuous function on \([a,b]\). The continuous F-transform is given by \[ f_i=\frac{\int_{a}^{b}A_{i}(x)f(x)dx}{\int_{a}^{b}A_{i}(x)dx}, \qquad i=1,\dots,k. \] The discrete F-transform is given by \[ f_i=\frac{\sum_{j=1}^{n}A_{i}(x_{j})f(x_{j})}{\sum_{j=1}^{n}A_{i}(x_{j})}, \] where \(x_{j}\in I\), \(j=1,\dots,n\) are given data \(\left( n\geq1\right) \) such that for each \(i\in\left\{ 1,\dots,k\right\} \) there exists \(p\in\left\{ 1,\dots,n\right\}\) with \(x_{p}\in\;\)supp\ \(A_{i}\), i.e., in the support of each atom of the partition we find at least one data point. The continuous inverse F-transform is \[ F_{k}(x)=\sum_{i=1}^{k}A_{i}\left( x\right) f_{i}\, , \] and the discrete inverse F-transform is \[ F_{n,k}(x)=\sum_{i=1}^{k} A_i(x)f_i \,. \] The authors consider the composition between the inverse and direct F-transform as an approximation operator and show, among other things, that if \(f\) is continuous, then there exists a sequence of fuzzy partitions with small support \(A_{m}\) such that the composition of the inverse and direct discrete F-transforms converges uniformly to \(f\). The authors then introduce B-splines, Shepard kernels, Bernstein basis polynomials based F-transforms, study their approximation properties, and finally conclude the paper with examples.