The purpose of this paper is to reduce the complexity of computing the components of the integral $F^m$-transform, $m\ge 0$, whose analytic expressions include definite integrals. We propose to use nontrivial quadrature rules with nonuniformly distributed integration points instead of the widely used Newton–Cotes formulas. As the weight function that determines orthogonality, we choose the generating function of the fuzzy partition associated with the $F^m$-transform. Taking into account this fact and the fact of exact integration of orthogonal polynomials, we obtain exact analytic expressions for the denominators of the components of the $F^m$-transformation and their approximate analytic expressions, which include only elementary arithmetic operations. This allows us to effectively estimate the components of the $F^m$-transformation for $0\le m\le 3$. As a side result, we obtain a new method of numerical integration, which can be recommended not only for continuous functions, but also for strongly oscillating functions. The advantage of the proposed calculation method is shown by examples.

The goal of this study is to solve fuzzy Fredholm integral equations of the second kind. In order to solve these equations with respect to fuzzy valued functions, we propose a very powerful and relatively simple technique called fuzzy transform. This approach allows the transformation of a fuzzy Fred-holm integral equation to a system of algebraic equations. A solution to this algebraic system gives the appropriate parameters of the inverse F1-transform. Hence, we can estimate the approximate solution to the original problem. The existence and uniqueness of the exact solution and approximate solution are also discussed.