On Higher Degree F-transforms Based on B-splines

s of MMA2015, May 26–29, 2015, Sigulda, Latvia c © 2015 ON HIGHER DEGREE F-TRANSFORMS BASED ON B-SPLINES MARTINS KOKAINIS and SVETLANA ASMUSS 1 Department of Mathematics, University of Latvia Zeļļu iela 8, R̄ıga, LV-1002, Latvia 2 Institute of Mathematics and Computer Science, University of Latvia Raiņa bulvāris 29, R̄ıga, LV-1459, Latvia E-mail: [email protected], [email protected] The concept of F -transforms (direct and inverse) was introduced in [1] and further extended to the higher degree F -transform [2] (so called F-transforms). The F-transforms have been successfully applied in image processing, data analysis and time series analysis during last years. We focus on approximation properties of the F-transform with respect to a generalized uniform fuzzy partition given by B-splines of odd degree. Let an interval [a, b] be fixed and k,N ∈ N. By A we denote the central B-spline of order 2k−1 with integer knots −k, . . . , k. Let h = (b−a)/(N+2k), ti = a + h (i + k), i = −k, . . . , N + k; then a = t−k < t0 < tN < tN+k = b. Functions A0, . . . , AN , where Ai(t) = A ((t− ti)/h), i = 0, . . . , N , form a (h, hk)-uniform generalized fuzzy partition of interval [a, b] in the sense of [3]. Our construction generalizes the commonly used uniform fuzzy partition given by a generating function of the triangular shape [2]. Our main result is as follows. Suppose that p is a polynomial, deg p ≤ 2k − 1 and m is a nonnegative integer s.t. 2m + 1 ≥ deg p. Then p coincides with its inverse F-transform (w.r.t. the (h, hk)-uniform generalized fuzzy partition based on B-splines of degree 2k− 1) F(p, ·) on interval [tk−1, tN−k+1]. Based on this result we obtain error estimations and prove that using B-splines improves the quality of approximation of a function by its inverse F-transform. If integers r,m satisfy 0 ≤ m ≤ k − 1, 1 ≤ r ≤ 2m + 2, then for f ∈ C[a, b] it holds that |f(x)−F(f, x)| ≤ O(h) for all x ∈ [tk−1, tN−k+1]. We consider the inverse F-transform in the context of local approximation methods given by splines and compare the obtained result with approximation properties of other local methods.