Approximation by multivariate higher degree F-transform based on B-splines

Modified multivariate F-transform based on B-splines

“…For a fixed degree the central B-spline is unique (up to a constant factor). The properties of B-splines and construction of a fuzzy partition using central B-spline as the generating function are described in more details in [7].…”

“…1 One could allow larger values of m by taking finer subgrids ∆,∆, see [7] for details; for simplicity, we consider here only the case when∆ consists only of the basic nodes.…”

“…In all these settings, properties of F -transforms significantly depend on basic functions which form a fuzzy partition. We focus on F mtransforms with respect to a spline-based fuzzy partition proposed in [6], generalized for the multivariate case in [7].…”

“…Previously we have proved that using B-splines may improve approximation properties of F m -transforms for smooth functions of one and also of many variables However, the approximation error estimations both in the univariate case [8] and in the multivariate case [7] are true only in the subset where the Ruspini condition is fulfilled. To eliminate this obstacle for the effective use of the advantages of the F m -transform technique in applications a modification of the spline-based F mtransform was proposed.…”