Uniform reconstruction of continuous functions with the RAFU method

Abstract

The RAFU (radical functions) method can be used to obtain the uniform reconstruction of a continuous function from its values at some of the points of partitions of a closed interval. In this work we will prove that we can reconstruct a continuous function from average samples of these points, from linear combinations of them and from local average samples given by convolution. A uniform error bound of order O h 3 2 ) + ω (h) with the step size h will be established. If these data are unknown but approximate values of them are known, uniform reconstruction will be also possible. Error estimates of order O h 3 2 + ω (h) + η with noise level η will be given. The case of a non-uniform net will be treated. Examples and algorithms will be also shown.

The taxi function

“…In robotics, for example, the concept of lane-keeping motion planning algorithms for mobile robots in order that the robot does not leave the lane if collision-free motion is available introduced in [9]. The road agency formation unit (RAFU) functions have been studied in Approximation Theory (the interested reader can see [10]- [17]). In this paper, the RAFU functions will be called the Taxi functions.…”

“…The proofs of Proposition 2 and Theorem 2 can be seen in [17]. According to these proofs, we define the functions E n (x) as in Proposition 1 but taking into account that k 1 = f (x 0 ) = f (x 1 ) and k i = f (x i ) for all i = 2, ..., n and then its corresponding Taxi functions C n (x) as in (4).…”